D.D. ANDERSON and J.R. JUETT
Communicated by Alexandru Zaharescu
Let R be a commutative ring with identity. For a, b ∈ R and n ≥ 1, an n- stage division chain is a sequence of equationsa=bq1+r1,b=r1q2+r2, . . . , rn−2=rn−1qn+rn. Ifrn= 0, we say thatbis ann-stage divisor ofa. The ring Risn-quasi-Euclidean (resp., quasi-Euclidean) if for nonzeroa, b∈R,bis ann- stage divisor ofa(resp.,bis anm-stage divisor ofafor somem≥1depending on aandb). We studyn-quasi-Euclidean rings and quasi-Euclidean rings and their relation to matrix reduction. We give a number of characterizations of quasi- Euclidean rings. Now,R is 2-quasi-Euclidean, precisely when it is Bezout with stable rank 1, or, equivalently, fora, b∈Rthere is anx∈Rwith(a, b) = (a+bx). We show that R is 3-quasi-Euclidean if fora, b ∈ R there existc, d ∈ R with (a, b) = (ca+ (1−cd)b)and give a similar characterization ofn-quasi-Euclidean rings. We characterize when the monoid ringR[X;S]or power series ringR[[X]]
is quasi-Euclidean.
AMS 2010 Subject Classication: 13A05, 13F05, 13F07, 13J05, 16S36.
Key words: Bezout ring, division chain, elementary divisor ring, Euclidean ring, Hermite ring, quasi-Euclidean ring, stable rank.
1. INTRODUCTION
Let R be a commutative ring with identity. For a, b∈ R and n ≥1, an n-stage division chain is a sequence of equations a= bq1 +r1, b =r1q2+r2, . . . , rn−2=rn−1qn+rn. Ifrn= 0, we say thatb is ann-stage divisor ofaor b n-dividesa, writtenb|na. The ringR isn-quasi-Euclidean ifb|nafor nonzero a, b ∈ R, and is quasi-Euclidean if for nonzero a, b ∈ R there is an m ≥ 1, depending on aand b, withb|ma.
The purpose of this paper is to study n-quasi-Euclidean rings and quasi- Euclidean rings and their relation to matrix reduction. In Section 2, we discuss matrix reduction and some rings related to it. In Section 3, we study n-quasi- Euclidean rings and quasi-Euclidean rings. Theorem 3.3 gives a number of characterizations of quasi-Euclidean rings. Now, R is 2-quasi-Euclidean, pre- cisely when it is Bezout with stable rank 1, or, equivalently, for a, b∈R there is anx ∈R with (a, b) = (a+bx). We show thatR is 3-quasi-Euclidean if for
REV. ROUMAINE MATH. PURES APPL. 58 (2013), 4, 417435
a, b ∈ R there exist c, d ∈R with (a, b) = (ca+ (1−cd)b) and give a similar characterization ofn-quasi-Euclidean rings (see Theorems 3.7 and 3.9). We end by characterizing when the monoid ring R[X;S]or power series ring R[[X]]is quasi-Euclidean (Theorems 3.16 and 3.19, respectively).
2. PRELIMINARIES
Throughout, all rings will be commutative with 16= 0. For a ring R, we will use R× to denote its group of units and abbreviateR∗ =R\ {0}.
For a ring R and n≥1, the general linear group of degree nis the group GLn(R) of invertible n×n matrices over R. The elementary row operations on a matrix over R are the operations of the following types: (1) multiplying a row by a unit, (2) adding a multiple of one row to another, and (3) trans- posing two rows. The elementary column operations are dened analogously.
Instead of applying elementary row (resp., column) operations, we can equiv- alently left-multiply (resp., right-multiply) by appropriate invertible matrices called elementary matrices. These are the matrices of the following types: (1) invertible diagonal matrices, (2) matrices called transvections that dier from the identity matrix by one nonzero element o the main diagonal, and (3) per- mutation matrices, i.e., matrices obtained by permuting the rows (equivalently, columns) of the identity matrix. Matrices of type (2) correspond to operations of type (2), and matrices of types (1) and (3) correspond to compositions of operations of types (1) and (3), respectively. Following [10], we callR a GEn- ring (the GE stands for generalized Euclidean) if GLn(R) is generated by the elementary matrices.
A ring is an elementary divisor ring if every matrix over it admits a diag- onal reduction, i.e., for every matrix A over it there are invertible matricesP and Q withP AQ=diag(a1, . . . , an) and each ai |ai+1. (A matrix (aij) is di- agonal ifaij = 0fori6=j; we do not require a diagonal matrix to be square.) A ring is Hermite if every matrix over it admits a lower trapezoidal reduction, i.e., for every matrix A there is an invertible matrix U withAU lower trapezoidal.
(A matrix (aij) is lower (resp., upper) trapezoidal if aij = 0 for i < j (resp., i > j). Hence, a matrix is diagonal if and only if it is both upper and lower trapezoidal. Some sources use triangular in place of trapezoidal, but we have avoided that terminology because a triangular matrix is usually required to be square.) By symmetry, we can equivalently change AU to U A while changing the lower to upper. If R is a Hermite ring, then we can further normalize these trapezoidal reductions so that the Hermite form is achieved.
Hermite rings can alternatively be characterized as the rings over which all1×2 (resp., 2×1) matrices admit a diagonal reduction ([19], Theorem 3.5), or as
rings R satisfying the property that for every a, b ∈ R there are d, a0, b0 ∈ R with a = da0, b = db0, and (a0, b0) = R ([15], Theorem 3). For a ring to be an elementary divisor ring, it is sucient for all1×2 (resp., 2×1) and2×2 matrices over it to admit a diagonal reduction ([19], Theorem 5.1), and, for an integral domain, one only need consider the 2×2matrices.
An elementary reduction ring is an elementary divisor ring where the reduction can be achieved by elementary row and column operations. Equiv- alently, an elementary reduction ring is a GE2 elementary divisor ring, or, equivalently, an elementary divisor ring that is GEn for all n ≥1. The anal- ogous strengthening of the Hermite denition, i.e., being able to achieve the lower (resp., upper) trapezoidal reduction with elementary column (resp., row) operations, characterizes the quasi-Euclidean rings that we will study in the next section.
The notion of stable rank has proven useful in the study of Hermite and elementary divisor rings. The stable rank of a ring R is the inmum sr(R) of the positive integers n such that (a1, . . . , an+1) =R implies that there are b1, . . . , bn ∈ R with (a1 +b1an+1, . . . , an +bnan+1) = R. It can be shown that this condition holds for n+ 1 if it holds for n. We dene the almost stable rank of R to be the supremum asr(R) of the stable ranks of its proper homomorphic images, or, equivalently ([3], Proposition 4), as the inmum of the positive integers n such that a1 6= 0 and (a1, . . . , an+2) = R implies that there are b2, . . . , bn+1 ∈ R with (a1, a2 +b2an+2, . . . , an+1 +bn+1an+2) = R. We note that asr(R)≤sr(R)≤asr(R) + 1. Hermite rings can be characterized as Bezout rings with stable rank at most 2 ([27], Theorem 1). (Recall that a ring is Bezout if every nitely generated ideal is principal.) Also, by ([22], Theorem 3.7), a Bezout ring with almost stable rank 1 is an elementary divisor ring, but [24] givesZ+XQ[X]as an example of an elementary divisor domain (in fact, an elementary reduction ring) with almost stable rank 2. It is easily shown that if I is an ideal contained in the Jacobson radical J(R) of R, then sr(R) =sr(R/I). (However, the inequality asr(R)≥asr(R/I)can be strict. For example, we have asr(Z[[X]]) = 2 and asr(Z[[X]]/J(Z[[X]])) = asr(Z) = 1.) From this observation, we immediately deduce two noteworthy consequences:
(A) IfR is a Bezout ring andI is an ideal contained in its Jacobson radical, thenR is Hermite if and only ifR/I is.
(B) A Bezout ring whose Jacobson radical contains a prime ideal is Hermite.
A ring Ris von Neumann regular if it satises one of the following equiv- alent conditions: (1) for every a ∈ R there is an x ∈ R with a2x = a, (2) every element is a unit times an idempotent, (3) every principal ideal is gen- erated by an idempotent, (4) every nitely generated ideal is a principal ideal
generated by an idempotent, or (5) R is zero-dimensional and reduced. In addition to semihereditary rings being arithmetical, we have the following im- plications: von Neumann regular ⇒ Bezout with stable rank 1 ⇒ elementary reduction ring⇒elementary divisor ring⇒Hermite⇒Bezout⇒arithmetical
⇒ Gaussian ⇒ Prufer, and none of the implications reverse. (Recall that a ring is arithmetical if its lattice of ideals is distributive, and is Prufer if every nitely generated regular ideal is invertible. Finally, the ring R is Gaussian if for f, g ∈ R[X] we have c(f g) = c(f)c(g), where c(f) denotes the content of f.) In the case of presimpliable rings (rings with all zero divisors contained in the Jacobson radical, or, equivalently, rings where xy =x ⇒ x = 0 or y is a unit), the Hermite and Bezout properties are equivalent ([19], Theorem 3.2) (this is generalized by observation (B) above), and the arithmetical, Gaussian, and Prufer properties are equivalent ([20], Theorem 64) (slightly augment the proof to obtain the presimpliable generalization). In the case of domains, we can add semihereditary to the second equivalence. It remains an open question whether there is a Bezout domain that is not an elementary divisor domain. We refer the reader to [17] for information on the aforementioned rings between Prufer and semihereditary rings, and to [19] for information on Bezout, Hermite, and elementary divisor rings.
3. DIVISION CHAINS
Let R be a ring. Following Cooke [11], fora, b∈R and n≥1, we dene ann-stage division chain starting with the pair (a, b)and ending withrn to be a sequence of equations (*):
a = bq1+r1, b = r1q2+r2, r1 = r2q3+r3,
. . .
rn−2 = rn−1qn+rn.
In this case, we will write DC(a, b, rn, n)or DC(a, b, rn). For convenience, we will sometimes take r−1 = a and r0 = b. A division chain is terminating if it ends with 0, and minimal if there is no shorter division chain with the same start and end. We say b is an n-stage divisor of a or b n-divides a, written b |n a, if DC(a, b,0, n), i.e., if there is a terminating n-stage division chain starting with (a, b). We dene length(a, b, r) = inf{n | DC(a, b, r, n)}
and length(a, b) = length(a, b,0). (Note that length(a, b, r) = ∞ if DC(a, b, r) does not hold.) We call b a strictn-stage divisor of aif length(a, b) =n <∞, or, equivalently, there is a minimal terminating n-stage division chain starting
with (a, b). We dene the diameter of R to be diam(R) = sup{length(a, b) | a, b∈R∗}= sup{length(a, b)|a∈R, b∈R∗}.
We next collect some simple facts about division chains and the|nrelation.
Theorem 3.1. Let R be a ring, a, b, d, r∈R, and n≥1. (1) DC(a, b, r,1)⇔a−r∈(b).
(2) If DC(a, b, r, n) as in (*), then DC(ri−1, ri, r, n−i) for 0≤i≤n−1.
(3) DC(a, b, r, n)⇒DC(a, b, r, n+1)and DC(b, a, r, n+1). Hence, length(b, a, r)
≤length(a, b, r) + 1.
(4) DC(a, b, r, n)⇒DC(ad, bd, rd, n).
(5) If dis a regular common divisor of aand b, then:
(a) DC(a, b, r, n)⇒DC(ad,bd,rd, n). (b) length(a, b, r) =
(length(ad,db,rd), d|r
∞, d-r.
(6) Let u, v ∈ R×. Then DC(a, b, r, n) ⇒ DC(ua, vb, ur, n) if n is odd and DC(ua, vb, vr, n) ifn is even. Hence, DC(a, b, r, n) ⇒DC(ua, vb, ur, n+ 1)and DC(ua, vb, vr, n+ 1).
(7) For a division chain as in (*), we have (a, b) = (b, r1) = (r2, r3) =· · ·= (rn−1, rn).
(8)
length(a, b, r) =
(1 + inf{length(b, x, r)|a−x∈(b)}, a−r /∈(b)
1, a−r∈(b).
Proof. Parts (1), (2), and (7) are clear, and part (8) follows from parts (1) and (2). We now prove the remaining parts.
(3) Assume DC(a, b, r, n) as in (*). Replacing the last equation in (*) with the two equations rn−2 =rn−1(qn+ 1) + (rn−rn−1) and rn−1 = (rn− rn−1)(−1) +rn, we see that DC(a, b, r, n+ 1). Insertingb=a·0 +bbefore (*), we see that DC(b, a, r, n+ 1).
(4) Multiply the equations in (*) by d.
(5) For part (a), divide the equations in (*) byd, noting thatddivides each ri. For part (b), the d | r case follows from (4) and (5a), while the d-r case follows from (7) and the fact that (a, b)⊆(d).
(6) In view of (3), it will suce to show the rst statement. Given DC(a, b, r, n) as in (*), we reach our desired conclusion by writingua= (vb)(uv−1q1) + ur1,vb= (ur1)(vu−1q2) +vr2,ur1= (vr2)(uv−1q3) +ur3, and continuing in this fashion.
Corollary 3.2. Let R be a ring, a, b, d, r ∈R, andn≥1. (1) a|1b⇔a|b.
(2) If b|na as in (*), then ri |n−(i+1) ri−1 for −1≤i≤n−1.
(3) b|na⇒a|n+1b and b|n+1 a. Hence, length(b, a)≤length(a, b) + 1. (4) (a) b|n0.
(b) 0|na⇔a= 0 or n≥2.
Hence, length(0, b) = 1 and length(a,0) = 2for a6= 0. (5) b|na⇒bd|nad.
(6) If b |n a and d is a regular common divisor of a and b, then bd |n ad. Hence, if d is a regular common divisor of a and b, then length(a, b) = length(ad,db).
(7) For u, v∈R×,a|nb⇔ua|nvb, and length(a, b) =length(ua, vb). (8) If b|naas in (*), then(a, b) = (rn−1). Moreover, in this case, if(a, b) =
(d), then there are a0, b0 ∈R with a=da0, b=db0, and (a0, b0) =R. (9) b |2 a if and only if there is an x ∈ R with (a+bx) = (a, b). Hence, if
(a, b) =R, then b|2a if and only if there is an x∈R with a+bx a unit.
(10)
length(a, b) =
(1 + inf{length(b, r)|a−r ∈(b)}, b-a
1, b|a.
Proof. We prove parts (8) and (9). The rest are immediate from Theo- rem 3.1.
(8) The rst statement is ([11], Proposition 3), but we can also immediately get it from Theorem 3.1 part (7). For the moreover statement, by ([15], Lemma 4) it will suce to consider the case d = rn−1. If n = 1, then a=rn−1q1,b=rn−1·1, and (q1,1) =R. So let us assumen≥2. Then (*) gives r1 |n−1 b, so by induction we have b =rn−1a0 and r1 = rn−1b0 for somea0, b0 ∈R with(a0, b0) =R. Thena=bq1+r1=rn−1(a0q1+b0), b=rn−1a0, and(a0q1+b0, a0) = (a0, b0) =R.
(9) (⇒): Ifb|2 aas in (*), then(a, b) = (r1) = (a−bq1). (⇐): If there is an x ∈R with (a+bx) = (a, b), then a=b(−x) + (a+bx) and a+bx |b, showing thatb|2 a.
For n≥1 and a functionφ from R into a well-ordered set W, we call R an n-stage (W-)Euclidean ring with respect to φif for every a, b∈R∗ there is an n-stage division chain as in (*) with φ(rn) < φ(b). A (W-)Euclidean ring is a 1-stage (W-)Euclidean ring. (Cooke [11] restricts the denitions to the case W = ω = {0,1,2, . . .}, so his (n-stage) Euclidean ring corresponds to
what we would call an (n-stage) ω-Euclidean ring. We have expanded the denition so that a 1-stage Euclidean ring corresponds to a Euclidean ring in the sense of Samuel [25].) By Corollary 3.2, one can equivalently replace the n-stage division chains with division chains of at mostnstages in the denition of n-stage (W-)Euclidean. Hence, n-stage (W-)Euclidean ⇒ (n+ 1)-stage (W-)Euclidean.
Let R be a ring. For n ≥1, we call R (strictly) n-quasi-Euclidean if for each a, b ∈ R∗ we have b |n a (and there are a0, b0 ∈ R∗ with b0 strictly n- dividinga0). Forn≥2, we can equivalently replace the R∗ with R in these denitions. Equivalently, an n-quasi-Euclidean ring is a ring with diameter at most n, and a strictly n-quasi-Euclidean ring is a ring with diameter n. We note that diam(R) = inf{n| R isn-quasi-Euclidean}. Of course, an n-quasi- Euclidean ring is n-stage ω-Euclidean with respect to the functionφ:R →ω given byφ(0) = 0and φ(a) = 1 fora6= 0.
Following Chen [9], we call a ring R stably Euclidean if length(a, b)<∞ for every comaximala, b∈R (equivalently, R∗), or, equivalently ([9], Theorem 11.1.2), if every 1×2 (resp., 2×1) matrix over R with comaximal entries admits a diagonal reduction by elementary column (resp., row) operations, or, equivalently ([9], Theorem 11.1.3), if R is GE2 and every 1×2 (resp., 2×1) matrix over R with comaximal entries can be completed to an invertible2×2 matrix.
Theorem 3.3. The following are equivalent for a ring R. (1) length(a, b)<∞ for every a, b∈R∗ (resp., a, b∈R)
(2) There is a well-ordered set W and a function φ :R → W such that for every a, b∈R∗ we have DC(a, b, r) for somer∈R with φ(r)< φ(b). (3) There is a function φ : R → ω such that for every a, b ∈ R∗ we have
DC(a, b, r) for somer ∈R with φ(r)< φ(b).
(4) There is a well-ordered set W and a function φ:R×R → W such that for everya, b∈R∗ there areq, r∈R witha=bq+r andφ(b, r)< φ(a, b). (5) There is a function φ:R×R→ω such that for every a∈R andb∈R∗
there are q, r∈R with a=bq+r andφ(b, r)< φ(a, b).
(6) Every 1×2 (resp., 2×1) matrix over R admits a diagonal reduction by elementary column (resp., row) operations.
(7) Every matrix over R admits a lower (resp., upper) trapezoidal reduction by elementary column (resp., row) operations.
(8) The ringR is Hermite and GE2.
(9) The ringR is Hermite and GEn for all n≥1. (10) The ringR is Hermite and stably Euclidean.
Proof. Note rst that the dierent forms of(6) and(7) are equivalent by symmetry, and the two forms of (1) are equivalent since length(a, b) is 1 or 2 when a or b is zero. We have(1) ⇔ (3) by ([11], Proposition 1), and a small adjustment of the proof gives (1) ⇔ (2). The equivalence (3) ⇔ (4) ⇔ (6) is from ([4], Theorem 5, Proposition 23). In ([11], Proposition 4) it is shown that (3) impliesR is GEn for alln≥1, and, since (6) clearly implies thatR is Hermite, we get(3)⇒(9). The equivalence(3)⇔(10)is ([9], Corollary 11.3.3).
Because (9) ⇒ (8)⇒ (6)⇐ (7) and (5)⇒ (4) are clear, and (10) ⇒ (7) can be shown by trivial adjustments to the proof of ([9], Corollary 11.2.5), all that remains is to show (1) ⇒ (5). Assume length(a, b) < ∞ for every a, b ∈ R∗. Dene φ:R×R →ω byφ(a,0) = 0and φ(a, b) =length(a, b) forb6= 0. Take any a∈R andb∈R∗. Ifb-a, then by Corollary 3.2 part (10) we have
φ(a, b) = 1 + inf{length(b, r)|a−r∈(b)}= 1 + min{φ(b, r)|a−r∈(b)}, so we may write a = bq+r for some q, r ∈ R with φ(b, r) < φ(a, b). On the other hand, if b | a, then a = b(ab) + 0 and φ(a, b) = length(a, b) = 1 > 0
=φ(b,0).
Following Bougaut [4], we call a ring R satisfying the equivalent condi- tions of Theorem 3.3 quasi-Euclidean. (There are other names for such a ring.
O'Meara [23] says that such a ring satises the Euclidean chain condition, and Cooke [11] calls such a ringω-stage Euclidean.) It is immediate from (6) or (7) that elementary reduction ring ⇒ quasi-Euclidean ⇒ Hermite, and from (2) we see thatn-quasi-Euclidean ⇒ n-stage Euclidean ⇒ quasi-Euclidean. How- ever, Bougaut [4] givesR[X, Y]/(X2+Y2+ 1)as an example of a PID (hence, elementary divisor domain) that is not quasi-Euclidean (in fact, is not stably Euclidean). Another example, noted by Cooke [11], is the ring of algebraic integers in Q(√
−19). An example of a stably Euclidean ring that is not quasi- Euclidean is a quasilocal ring that is not Bezout. The paper [8] constructs examples of quasi-Euclidean domain that are not 2-stage Euclidean, such as Z+XQ[X]and K[X] +Y K(X)[Y], whereK is a nite eld.
It is well known that a(n) (ω-)Euclidean domain is a PID, and ([14], Theorem 5.3) shows that the converse is true if the stable rank is 1. However, there are examples of ω-Euclidean domains with stable rank 2, such as Z or K[X], whereK is a eld.
Theorem 3.4. Let R be a ring and φ : R×R → ω. The following are equivalent.
(1) The ringR is quasi-Euclidean andφ is the unique smallest function sat- isfying Theorem 3.3(5).
(2) (a) φ(a, b) = 0⇔b= 0, and
(b) fora∈R andb∈R∗we have φ(a, b) = 1+min{φ(b, r)|a−r ∈(b)}.
(3) The ringR is quasi-Euclidean,φ(a,0) = 0, and φ(a, b) =length(a, b) for b6= 0.
In this case, we have diam(R) =|φ(R×R∗)|.
Proof. (3)⇒(2): See the proof of (1)⇒(5) in Theorem 3.3.
(2) ⇒ (1): Assume (2) and let ψ : R×R → ω be any function as in Theorem 3.3(5). Take anya, b∈R. Ifψ(a, b) = 0, then Theorem 3.3(5) implies that b = 0, and hence, φ(a, b) = 0 = ψ(a, b). So let us assume ψ(a, b) ≥ 1. If b = 0, then φ(a, b) = 0 ≤ ψ(a, b), so let us assume b 6= 0. Then there are q, r ∈ R with a = bq +r and ψ(b, r) < ψ(a, b). By induction, we have φ(b, r)≤ψ(b, r), soψ(a, b)≥ψ(b, r) + 1≥φ(b, r) + 1≥φ(a, b).
(1)⇒(3): Follows from (3)⇒ (1) and uniqueness.
The last statement is clear from (3) and the observation that (2) implies thatφ(R×R)is an interval.
Theorem 3.5.
(1) An n-quasi-Euclidean ring is m-quasi-Euclidean for m≥n. (2) A ring is 1-quasi-Euclidean if and only if it is a eld.
(3) The following are equivalent for a ring R. (a) R is 2-quasi-Euclidean.
(b) R is Bezout with stable rank 1.
(c) For everya, b∈R, there is an x∈R with (a, b) = (a+bx).
(4) The following are equivalent for a ring R and a nonempty family of alge- braically independent indeterminates {Xλ}λ∈Λ.
(a) R is arithmetical.
(b) R({Xλ}) is arithmetical.
(c) R({Xλ}) is Bezout with stable rank 1.
(5) A principal ideal ring with stable rank 1 is Euclidean.
Proof.
(1) Follows from Corollary 3.2 part (3).
(2) By Corollary 3.2 part (1), a ringRis 1-quasi-Euclidean if and only ifa|b for every a, b∈R∗. The result is now clear.
(3) (We note that (b) ⇔ (c) for domains is ([14], Proposition 5.1).) (a)⇒ (c): Corollary 3.2 part (9).
(c)⇒ (b): Clear.
(b) ⇒ (a): AssumeR is Bezout with stable rank 1. Take any a, b∈R∗. SinceRis Hermite, we may writea=a0dandb=b0d, where(a0, b0) =R. Since sr(R) = 1, there is an x∈R with(a0+b0x) =R. So, by Corollary 3.2 part (9), we obtainb0 |2 a0, and from Corollary 3.2 part (5) we deduce b|2 a.
(4) We have (a)⇔(b)⇔R({Xλ})is Bezout by ([1], Theorem 8(1)), and (c)
⇒ (b) is clear. So it will suce to show thatR({Xλ}) has stable rank 1.
Assume(fh,gh) =R({Xλ}), wheref, g, h∈R[{Xλ}]withc(h) =R. Then R({Xλ}) = (f, g), so R({Xλ}) = c(f)R({Xλ}) +c(g)R({Xλ}) = c(f + XαNg)R({Xλ})forN large enough, whereα∈Λ. Thereforec(f+XαNg) = R, and hence f +XαNg is a unit in R({Xλ}). Therefore fh + (gh)XαN =
1
h(f +XαNg) is a unit, as desired.
(5) Recall that a PIR is a nite direct product of PID's and SPIR's (where a special principal ideal ring or SPIR is a local Artinian PIR). Of course, a SPIR is Euclidean, and a PID with stable rank 1 is Euclidean by ([14], Theorem 5.3), so the result follows from the fact that a nite direct prod- uct of Euclidean rings is Euclidean.
Theorem 3.5 part (3) gives us several examples of 2-quasi-Euclidean rings, such as semi-quasilocal Bezout rings, zero-dimensional Bezout rings, or the rings R({Xλ}) where R is arithmetical. However, recall that a PID (hence, a one-dimensional Bezout ring) need not be quasi-Euclidean. The following construction gives a 2-quasi-Euclidean ring with an arbitrary lattice-ordered group as its group of divisibility.
Example 3.6. (Every lattice-ordered abelian group is the group of di- visibility of a 2-quasi-Euclidean domain, i.e., a Bezout domain with stable rank 1.) Let (G,≤) be a lattice-ordered abelian group, K be a eld, and K[X;G] be the group ring of G over K. Dene w : K[X;G] → G∪ {∞}
by w(0) = ∞ and w(Pn
i=1aixgi) = inf{gi}ni=1. Then w extends to a semi- valuation w : L → G∪ {∞}, where L is the quotient eld of K[X;G]. Let D = w−1(G+∪ {∞}). Heinzer [18] has shown that D is Bezout with stable rank 1.
We note that having stable rank 1 is not an ideal-theoretical property. Let R be a Bezout domain with quotient eldK. So its group of divisibility G= K×/R× is a lattice-ordered group. Let D be the Bezout domain constructed in Example 3.6; so D has stable rank 1. Let L(R) and L(D) be the lattice of ideals of R and D, respectively. Then the map θ :L(R) → L(D) given by θ(0) = 0andθ(I) = (XiR× |i∈I\{0})is a multiplicative lattice isomorphism.
Here,D has stable rank 1 whileR need not.
The characterizations of 2-quasi-Euclidean rings given in Theorem 3.5 part (3) have an analog for n-quasi-Euclidean rings for n ≥ 3. We next explicitly handle the case n= 3 and then formulate the general case.
Theorem 3.7. The following are equivalent for a ring R.
(1) R is 3-quasi-Euclidean.
(2) For a, b∈R, there exist c, d∈R with (a, b) = (ca+ (1−cd)b).
(3) R is Hermite and (2) holds for comaximal a, b∈R.
Proof. (1) ⇒ (3): Let R be 3-quasi-Euclidean and a, b ∈ R. Then R is Hermite, and we can write a = bq+r and b = rq0 +r0, where r0 | r. Then r0 =b−rq0 =b−(a−bq)q0 =−q0a+(1+q0q)b, so(a, b) = (r0) = (ca+(1−cd)b), wherec=−q0 and d=q.
(3) ⇒ (2): Assume (3) and let a, b ∈ R. Since R is Hermite, there are a0, b0, x∈R witha=a0x,b=b0x, and(a0, b0) =R. By (3), there are c, d∈R with(a0, b0) = (ca0+ (1−cd)b0), and hence (a, b) = (ca+ (1−cd)b).
(2) ⇒ (1): Assume (2) and let a, b ∈ R. Then there are c, d ∈ R with (a, b) = (ca+ (1−cd)b). Let r2 = ca+ (1−cd)b and r1 = a−bd. Then a=bd+r1,b=bcd−ca+r2 =r1(−c) +r2, andr2|r1, sob|3a.
We now extend Theorem 3.7 to the general case. We recursively dene a sequence of polynomials {αn}∞n=0 ⊆ Z[X1, X2, . . .] by α0 = 0, α1 = 1, and αn=αn−2−Xn−1αn−1 for n≥2. It is easily seen thatαn∈Z[X1, . . . , Xn−1] and degαn=n−1for n≥1.
Lemma 3.8. LetR be a ring, a, b∈R, and n≥1. Thenb|naif and only if we can write (a, b) = (αn−1(c1, . . . , cn−2)a+αn(c1, . . . , cn−1)b).
Proof. We have already established the casesn= 1andn= 2, so we may assumen≥3.
(⇒): Assumeb|na. Then there areq, r∈Rwitha=bq+r andr |n−1 b. By induction, there are c1, . . . , cn−2 ∈ R with (b, r) = (αn−2(c1, . . . , cn−3)b+ αn−1(c1, . . . , cn−2)r),and thus:
(a, b) = (b, r)
= (αn−1(c1, . . . , cn−2)a+ (αn−2−qαn−1)(c1, . . . , cn−2)b)
= (αn−1(c1, . . . , cn−2)a+αn(c1, . . . , cn−2, q)b).
(⇐): Assume we can write(a,b) = (αn−1(c1, . . . , cn−2)a+αn(c1, . . . , cn−1)b). Then:
(b, a−bcn−1) = (a, b)
= (αn−1(c1, . . . , cn−2)a+ (αn−2−cn−1αn−1)(c1, . . . , cn−2)b)
= (αn−2(c1, . . . , cn−3)b+αn−1(c1, . . . , cn−2)(a−bcn−1)),
so, by induction we have a−bcn−1 |n−1 b. But a = bcn−1+ (a−bcn−1), so b|na.
Theorem 3.9. The following are equivalent for a ring R and n≥2.
(1) R is n-quasi-Euclidean.
(2) For a, b∈R, there are c1, . . . , cn−1 ∈R with
(a, b) = (αn−1(c1, . . . , cn−2)a+αn(c1, . . . , cn−1)b).
(3) R is Hermite and (2) holds for comaximal a, b∈R.
Proof. We have (1)⇔(2)by Lemma 3.8, and it is clear that (1)and (2) together imply (3), so it will suce to show(3)⇒(2). Assume (3) and leta, b∈ R. SinceRis Hermite, there area0, b0, d∈Rwitha=a0d,b=b0d, and(a0, b0) = R. By (3), there are c1, . . . , cn−1 ∈ R with (a0, b0) = (αn−1(c1, . . . , cn−2)a0+ αn(c1, . . . , cn−1)b0), and multiplying bydnishes the proof.
Proposition 3.10. Homomorphic images, localizations, and overrings of (n-)quasi-Euclidean rings are (n-)quasi-Euclidean. Furthermore, when pass- ing from a ring to a homomorphic image or localization, the diameter is not increased.
Proof. The last statement will follow from the proof of the rst. Overrings of Bezout rings are localizations, so we only need to show the homomorphic image and localization parts. The former is easily shown by applying a given homomorphism to the equations in (*). We will demonstrate the latter by showing that, given a ring R, a multiplicatively closed subsetS of R,s, t∈S, and a, b∈Rwitha|m binR, we have as |m bt inRS. In this case, we certainly have a1 |m b1 inRS, and we can then apply Corollary 3.2 part (7) to reach the desired conclusion.
Proposition 3.11. Let {Rλ}λ∈Λ be a nonempty family of rings.
(1) If |Λ| ≥2, then diam(Q
Rλ) = supλmax(diam(Rλ),2). Hence, if n≥2 and each Rλ is n-quasi-Euclidean, then Q
Rλ isn-quasi-Euclidean.
(2) IfQ
Rλ isn-quasi-Euclidean (resp., quasi-Euclidean), then so is eachRλ. Moreover, if Q
Rλ is strictly n-quasi-Euclidean, then at least one Rλ is strictly n-quasi-Euclidean.
(3) If Λ is nite, then Q
Rλ is quasi-Euclidean if and only if each Rλ is.
Proof. Follows from the denitions applied to each coordinate, or from a suitable use of Theorem 3.4. We leave the details to the reader.
Lemma 3.12. Let R be a ring and I be an ideal contained in its Jacobson radical. The following are equivalent for comaximal a, b∈R.
(1) b|na in R.
(2) b+I |na+I inR/I.
(3) b+x|na+y for everyx, y∈I.
Proof. We abbreviateR¯ =R/I and x¯=x+I forx∈R. The implication (3)⇒(1)⇒(2)is clear. Now assume¯b|n¯a. Ifn= 1, thenR¯¯b= ¯R¯a+ ¯R¯b= ¯R, so Rb = R and hence, b+x is a unit for every x ∈ I. So let us assume n ≥ 2. Then we may write ¯a = ¯bq¯+ ¯r for some q, r ∈ R with ¯r |n−1 ¯b. Write a = bq +r +z for some z ∈ I. Then, for every x, y ∈ I we have a+y = (b+x)q+ (r+ (y+z−qx)), andr+ (y+z−qx)|n−1b+xby induction, and thus b+x|na+y.
McGovern ([22], Corollary 2.3) shows that a Bezout ringR is an elemen- tary divisor ring if and only if R/J(R) is. (Examining the proof shows that J(R) can actually be replaced with any ideal I ⊆ J(R).) We have analo- gous results for stably Euclidean rings, quasi-Euclidean rings, and elementary reduction rings.
Corollary 3.13. Let R be a ring and I be an ideal contained in its Jacobson radical. Then R is stably Euclidean if and only if R/I is.
Theorem 3.14. Let Rbe a Bezout ring and I be an ideal contained in its Jacobson radical.
(1) The ringR is quasi-Euclidean if and only if R/I is.
(2) diam(R/I)≤diam(R)≤max(2,diam(R/I)). Proof.
(1) This is immediate from Corollary 3.13 and the fact thatR is Hermite if and only if R/I is, but we will also give an alternate proof in order to facilitate the proof of part (2).
(⇒): Proposition 3.10.
(⇐): Assume R/I is quasi-Euclidean. Take any a, b ∈ R. Since R/I is Hermite, the same holds for R, so there are a0, b0, d ∈ R with a = da0, b=db0, and(a0, b0) =R. We get
length(a, b)≤length(a0, b0) =length(a0+I, b0+I)<∞ by Corollary 3.2 part (5) and Lemma 3.12.
(2) We get diam(R) ≥ diam(R/I) by Proposition 3.10, and from the proof of part (1) we obtain diam(R)≤max(2,diam(R/I)).
Corollary 3.15. Let R be a Bezout ring and I be an ideal contained in its Jacobson radical. Then R is an elementary reduction ring if and only if R/I is.
Proof. (⇒): Homomorphic images of elementary reduction rings are ele- mentary reduction rings.
(⇐): Since elementary reduction rings are precisely the quasi-Euclidean elementary divisor rings, this follows from Theorem 3.14 and the aforementio- ned result thatR is an elementary divisor domain if and only if R/I is.
By a monoid, we mean an additive commutative semigroup with identity.
We denote the group of units of a monoid S by S×, and we abbreviate S∗ = S\{0}. Given a ringRand a monoidS, we construct the monoid ringR[X;S] = {a1Xs1 +· · ·+anXsn | a1, . . . , an ∈ R, s1, . . . , sn ∈ S} with addition and multiplication dened analogously to polynomial addition and multiplication.
For example, we have R[X;Z+] =R[X]and R[X;Z] = R[X, X−1]. A monoid S is torsion-free ifns =nt⇒s=t for n≥1 and s, t∈S, and cancellative if a+b=a+c ⇒b=c for a, b, c ∈S. A monoid is Prufer if it is an ascending union of cyclic submonoids, where a monoid is cyclic if it is generated (as a monoid) by a single element. An excellent resource for monoid rings is [16].
The article [5] has a discussion on the history of the following theorem in the special case of polynomial rings, i.e., whenS =Z+.
Theorem 3.16. The following are equivalent for a ring R and a nonzero torsion-free cancellative monoid S.
(1) R is von Neumann regular and S is (up to isomorphism) a subgroup ofQ or a Prufer submonoid of Q+.
(2) Every 2-generated regular ideal of R[X;S]is invertible.
(3) R[X;S] is Prufer.
(4) R[X;S] is Gaussian.
(5) R[X;S] is arithmetical.
(6) R[X;S] is semihereditary.
(7) R[X;S] is Bezout.
(8) R[X;S] is Hermite.
(9) R[X;S] is quasi-Euclidean.
(10) R[X;S] is an elementary divisor ring.
(11) R[X;S] is an elementary reduction ring.
(12) R[X;S] is one-dimensional and reduced, and S is a group or Prufer monoid.
Proof. The equivalence (1) ⇔ (3) ⇔ (5) ⇔ (7) is given in ([16], Theo- rem 18.9), and of course (6)⇒(5)and (11)⇒ · · · ⇒(7)⇒(5)⇒ · · · ⇒(2)is true with any ring in place ofR[X;S], so it will suce to show(2)⇒(1)⇒(11) and(6)⇐(1)⇔(12). (The original version of ([16], Theorem 18.9) states that
S is a subgroup or Prufer submonoid of Qin (1), but this is equivalent to the way we have stated it since ([16], Theorem 2.9) asserts that a submonoid ofQ containing both positive and negative elements is a group.)
(2)⇒(1): In ([16], Theorem 18.9), it is shown that (1) holds if(f, g)2= (f2, g2) for every f, g ∈ R[X;S] with f or g regular. This condition holds if every 2-generated regular ideal ofR[X;S]is invertible. (See ([5], Remark 10).) (1)⇒(11): Assume (1). Two dierent proofs thatR[X]is an elementary divisor ring are given in ([26], Theorem) and ([6], Theorem 2). Slightly adapting the proof of the latter shows thatR[X]is in fact an elementary reduction ring.
(An alternate way to do this would be to combine ([6], Theorem 2) or ([26], Theorem) with ([12], Theorem 1.3), where the latter asserts that a polynomial ring is GE2 if and only if the base ring is zero-dimensional. Since the proof of the latter result is relatively dicult, a more elementary method would be to replace ([12], Theorem 1.3) by Proposition 3.18 below, which shows that a polynomial ring over a von Neumann regular ring is quasi-Euclidean.) The result now follows after we make three simple observations: (i) the property of being an elementary reduction ring is preserved by localization, (ii) an ascending union of elementary reduction rings is an elementary reduction ring, and (iii) the hypotheses on S ensure that R[X;S] is an ascending union of rings that are isomorphic to R[X]or R[X, X−1].
(1) ⇒ (6): Assume (1). We recall that Endo's Theorem [13] shows that semihereditary rings are precisely those arithmetical rings whose total quotient rings are von Neumann regular, so, in view of (1)⇔(5), all that remains is to show that T(R[X;S]) is von Neumann regular. The hypotheses on S ensure that we may write it as an ascending union of submonoids S = S
λ∈ΛSλ, where each Sλ is isomorphic to Z or Z+. By ([16], Corollary 8.6), the zero divisors of eachR[X;Sλ]are precisely those elements whose coecients are all annihilated by a single element. So an element of R[X;Sλ] is regular if and only if it is regular as an element ofR[X;S]. It now follows thatT(R[X;S]) = S
λ∈ΛT(R[X;Sλ]), where each of the latter total quotient rings is isomorphic to T(R[X]) =T(R[X, X−1]). Because McCarthy [21] has shown that R[X]is semihereditary, each of these is von Neumann regular, and thus T(R[X;S])is von Neumann regular.
(1)⇔(12): From ([16], Theorems 17.1 and 21.4] we obtaindimR[X;S] = dimR[X;G] = dimR[X1, . . . , Xn], where G is the quotient group of S and n is the torsion-free rank of G, i.e., the dimension of GZ∗ as a vector space over Q. Note that a torsion-free group has torsion-free rank 1 if and only if it is (isomorphic to) a subgroup of Q. Therefore R[X;S] is one-dimensional
⇔ R is zero-dimensional and n = 1 ⇔ R is zero-dimensional and S is (up to isomorphism) a submonoid of Q. By ([16], Theorem 9.17), we see that
R[X;S] is reduced if and only if R is. Since von Neumann regular rings are precisely the zero-dimensional reduced rings, combining the last two sentences gives (1)⇔(12).
The proof of (1) ⇔ (12) in Theorem 3.16 shows that, if R is a ring and S is a torsion-free cancellative monoid, then R[X;S] is one-dimensional (and reduced) if and only if R is zero-dimensional (von Neumann regular) and S is isomorphic to a submonoid of Q.
We state Theorem 3.16 again for the special case of domains, where we can add one more equivalent statement. Note that the hypotheses onDandS at the start of the following corollary are precisely what is required forD[X;S]
to be an integral domain ([16], Theorem 8.1).
Corollary 3.17. The following are equivalent for an integral domain D and a nonzero torsion-free cancellative monoid S.
(1) D is a eld and S is (up to isomorphism) a subgroup of Q or a Prufer submonoid of Q+.
(2) D[X;S]is Prufer.
(3) D[X;S]is Bezout.
(4) D[X;S]is quasi-Euclidean.
(5) D[X;S]is an elementary divisor domain.
(6) D[X;S]is an elementary reduction domain.
(7) D[X;S]is one-dimensional and S is a group or Prufer monoid.
(8) D[X;S]is Bezout with almost stable rank 1.
Proof. Since (8) ⇒ (2) is clear, and (1)(7) are equivalent by Theo- rem 3.16, we only need to show(1)⇒ (8). Assume (1). The hypotheses onS ensure that D is an ascending union of rings that are isomorphic to D[X] or D[X, X−1]. Each of these is a PID (in fact anω-Euclidean domain), and thus, has almost stable rank 1. (In [3] it is shown that, forn≥1, an n-dimensional Noetherian domain has almost stable rank at most n.) With the observation that an ascending union of rings each having almost stable rank 1 has almost stable rank 1, we nish the proof.
Note that the analog of (8) above cannot be added to Theorem 3.16 since sr(R) = asr(R) for a ring R with zero divisors by ([3], Theorem 3), but a polynomial ring never has stable rank 1.
As alluded to in the proof of Theorem 3.16, there is a direct proof that a ring Ris von Neumann regular if and only if R[X]is quasi-Euclidean. In fact, we have the following slightly stronger result that has some interest in its own right.
Proposition 3.18. Let R be a von Neumann regular ring, f, g ∈ R[X], and n= max(degg,0). Then g|n+2f.
Proof. First, we consider the case where n= 0. Then g∈ R, so there is an s∈ R withg2s =g. Let r1 = g+ (1−gs)f and q1 = (f s−1)sr1. Then gsr1=gand gq1+r1=g(f s−1) +g+ (1−gs)f =f, so g|2 f. Now assume n ≥ 1. Let b be the leading coecient of g, and pick s ∈ R with b2s = b.
Since deg (1−bs)g < n, we have(1−bs)g|n+1(1−bs)f by induction. Hence, (1−bs)g|n+2(1−bs)f, say:
(1−bs)f = (1−bs)gq1+r1, (1−bs)g = r1q2+r2,
r1 = r2q3+r3, . . .
rn = rn+1qn+2+rn+2,
wherern+2 = 0. The leading coecient of bsg isb, which is a unit inbsR[X], so there areq, r∈bsR[X]withbsf = (bsg)q+r and degr < degbsg ≤n. By induction, we haver |n+1 bsg inR[X], sobsg|n+2 bsf, say:
bsf = (bsg)q10 +r01, bsg = r10q02+r20,
r01 = r20q3+r30, . . .
r0n = rn+10 qn+20 +rn+20 ,
where r0n+2 = 0. Note that 1−bs(resp., bs) divides each ri (resp., r0i), and if necessary, we can modify our construction so that1−bs(resp.,bs) divides each qi (resp., qi0). Hence, eachriqj0 =r0iqj = 0. Deneq00k=qk+q0kandrk00=rk+rk0 for k =−1, . . . , n+ 2, where r−1 = (1−bs)f, r0 = (1−bs)g,r0−1 =bsf, and r00=bsg. Then fork=−1, . . . , n we have:
r00k+1qk+200 +rk+200 = (rk+1+r0k+1)(qk+2+qk+20 ) + (rk+2+rk+20 )
= (rk+1qk+2+rk+2) + (rk+10 qk+20 +r0k+2)
= rk+rk0
= rk00.
Since r00−1 = (1−bs)f+bsf =f, r000 = (1−bs)g+bsg =g, and rn+200 = 0 + 0 = 0, this shows thatg|n+2 f.
Theorem 3.19. The following are equivalent for a ring R.
(1) R is von Neumann regular and ℵ0-algebraically compact.
(2) R[[X]]is Gaussian.
(3) R[[X]]is arithmetical.
(4) R[[X]]is Bezout.
(5) R[[X]]is Hermite.
(6) R[[X]]is quasi-Euclidean.
(7) R[[X]]is an elementary divisor ring.
(8) R[[X]]is an elementary reduction ring.
(9) R[[X]]is Bezout with stable rank 1.
Proof. Because (9) ⇒ · · · ⇒ (2) holds with any ring in place of R[[X]], and since ([2], Theorem 17) gives the equivalence of (1)−(4), it will suce to show (1)⇒ sr(R) = 1. But any von Neumann regular ring has stable rank 1, and sr(R[[X]]) = sr(R) since (X) ⊆ J(R[[X]]) and R[[X]]/(X) ∼= R, so the proof is complete.
We note that ([7], Example 2) shows that the statement R[[X]] is semi- hereditary is not equivalent to the above statements.
Corollary 3.20. The following are equivalent for an integral domainD. (1) D is a eld.
(2) D[[X]] is Prufer.
(3) D[[X]] is Bezout.
(4) D[[X]] is quasi-Euclidean.
(5) D[[X]] is an elementary divisor domain.
(6) D[[X]] is an elementary reduction domain.
(7) D[[X]] is a discrete valuation ring.
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Received 11 October 2012 University of Iowa,
Department of Mathematics, Iowa City, IA 52242, U.S.A.
dan-anderson@uiowa.edu University of Iowa, Department of Mathematics, Iowa City, IA 52242, U.S.A.
jason-juett@uiowa.edu
