Gaussian polynomials and content ideal in pullbacks

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Gaussian Polynomials and Content Ideal in Pullbacks

Chahrazade Bakkari ^a & Najib Mahdou ^a

a Department of Mathematics, Faculty of Sciences and Technology Fès-Saïss , University S. M. Ben Abdellah , Fès, Morocco

Published online: 02 Feb 2007.

To cite this article: Chahrazade Bakkari & Najib Mahdou (2006) Gaussian Polynomials and Content Ideal in Pullbacks, Communications in Algebra, 34:8, 2727-2732, DOI: 10.1080/00927870600651695 To link to this article: http://dx.doi.org/10.1080/00927870600651695

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GAUSSIAN POLYNOMIALS AND CONTENT IDEAL IN PULLBACKS

Chahrazade Bakkari and Najib Mahdou

Department of Mathematics, Faculty of Sciences and Technology Fès-Saïss, University S. M. Ben Abdellah, Fès, Morocco

This article deals mainly with rings (with zerodivisors) in which regular Gaussian polynomials have locally principal contents. Precisely, we show that ifT Mis a local ring which is not a field,Dis a subring ofT/Msuch thatqfD=T/M,h T→T/M is the canonical surjection and R=h⁻¹D, then if T satisfies the property “every regular Gaussian polynomial has locally principal content,” then also R verifies the same property. We also show that ifDis a Prüfer domain andTsatisfies the property

“every Gaussian polynomial has locally principal content”, then R satisﬁes the same property. The article includes a brief discussion of the scopes and limits of our result.

Key Words: Content ideal; Gaussian polynomials; Pullbacks; Trivial ring extensions.

Mathematics Subject Classiﬁcation: Primary 13Agg; Secondary 13D05, 13B02.

1. INTRODUCTION

Let Rbe a commutative ring. We say that an ideal is regular if it contains a regular element, i.e., a non-zerodivisor. We say thatRhas locally the property P if each localization ofRat a maximal ideal has the property P.

The content Cf of a polynomial f ∈RX is the ideal of R generated by the coefﬁcients off. One of its properties is thatCis semi-multiplicative, that is, Cfg⊆Cf Cg. A polynomialf ∈RXis called aGaussian polynomialifCfg= Cf Cg for any polynomial g∈RX. A polynomial f ∈RX is a Gaussian polynomial provided Cf is a locally principal ideal (Heinzer and Huneke, 1997, Remark 1.1). Our guiding question is the converse of this property.

Question. Let R be a commutative ring and let f ∈RX be a Gaussian polynomial. IsCf a locally principal ideal ofR?

Some versions of this question appear in Tsang (1965), Anderson and Kang (1996), Glaz and Vasconcelos (1998), and Heinzer and Huneke (1997). We should

Received March 17, 2005; Revised November 3, 2005. Communicated by I. Swanson.

Address correspondence to Najib Mahdou, Department of Mathematics, Faculty of Sciences and Techniques Fès-Saïss, University S. M. Ben Abdellah, B.P. 2202, Fès, Morocco; Fax: 055-608214;

E-mail: mahdou@hotmail.com

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2728 BAKKARI AND MAHDOU

mention that without any hypotheses on the ring R or on the coefficients of the Gaussian polynomial f, this question has a negative answer (see Heinzer and Huneke, 1997, Remark 1.6). Heinzer and Huneke (1997, Theorem 3.3) proved that a locally Noetherian ring has the following property: every regular Gaussian polynomial has locally principal content ideal. Also, in Loper and Roitman (2004), the authors proved that a ring R which is locally a domain satisfies the same property (since a finitely generated regular ideal is invertible if and only if it is locally principal, (see Heinzer and Huneke, 1997, Remark 1.1).

The goal of this work is to exhibit a class of rings (with zerodivisors) that are neither locally Noetherian, nor locally a domain and satisfying the property that every regular Gaussian polynomial has locally principal content ideal. For this purpose, we show that if T M is a local ring which is not a field, D is a subring of T/M such that qfD=T/M, h T →T/M is the canonical surjection andR=h⁻¹D, then ifT satisfies the property stating that every regular Gaussian polynomial has locally principal content, then also R verifies the same property (Theorem 2.1(1)). We also show that if D is a Prüfer domain and T satisfies the property “every Gaussian polynomial has locally principal content ideal,” then R satisfies the same property (Theorem 2.1(2)). We also provide a counterexample showing that, without the assumptionqfD=T/M, Theorem 2.1 is not true even if Dis a field (Example 2.3). Finally, we exhibit a non-Noetherian local rings with zerodivisors where every regular Gaussian polynomial has principal content ideal (Examples 2.4 and 2.5).

2. MAIN RESULTS

In this section, we examine the possible transfer of the property “every (regular) Gaussian polynomial has locally principal content ideal” to pullbacks issued from a local ring, in order to provide some original examples satisfying this property.

Throughout this section, we adopt the following riding assumptions and notations: T M is a local ring, h T →T/M is the canonical surjection, D is a subring of T/M such that qfD=T/M, and R=h⁻¹D. It is easy to see that R⊆R_M ⊆T (sinceT Mis a local ring) andMR_M =MT =M. SinceD=R/M ⊆ R_M/M ⊆T/M,qfD=T/MandR_M/Mis a ﬁeld, thenT=R_M; in particularTisR- ﬂat. For more details on the property of such pullbacks, see Cahen (1988), Fontana (1980), and Gabelli and Houston (1997).

Theorem 2.1. LetT M D, andRas above. Then:

1) IfT satisﬁes the property “every regular Gaussian polynomial has locally principal content ideal,” then Rveriﬁes the same property.

2) Assume that D is a Prüfer domain. If T satisﬁes the property “every Gaussian polynomial has locally principal content ideal,” thenRsatisﬁes the same property.

The proof of this Theorem requires the following lemma.

Lemma 2.2. LetT M,D, andRas in Theorem 2.1,f a Gaussian polynomial ofRX, and suppose that either(i)T satisﬁes the property “every regular Gaussian polynomial has principal content ideal” and f is regular or (ii) T satisﬁes the property “every

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Gaussian polynomial has principal content ideal.” Then there existsx∈Tandg∈RX such that:

1) Cf T =xT;

2) f =xgandCf =xCg;

3) CgT =T.

Proof. 1) Let f =n

i=0a_iXⁱ be a Gaussian polynomial of RX, where n is a positive integer. Then f is also a Gaussian polynomial of T X since T= R_M. On the other hand, f regular in RX implies that it is regular in T X.

Therefore, sinceT Mis local and T isR-ﬂat, by hypothesis it follows thatCf T =Cf ⊗RT =xT for somex∈T.

2) We wish to construct a polynomialg∈RXsuch thatf =xgandCf = xCg. Two cases are then possible.

Case 1. xM. In this case, xis invertible in T since T M is a local ring and soCf T =xT =T. Hence, we may assume thatx=1 and it sufﬁces to take g=f.

Case 2. x∈M. Hence, a_i∈Cf ⊆Cf T =xT for each i=0 n. Then there existsb_i∈Rands_i∈R\M such thata_i=xb_i/s_i. Thus forx=x/_n

i=0s_j∈ T, we have a_i=xa_i, where a_i=n

j=0j=is_jb_i∈R. For g=n

i=0a_iXⁱ∈RX, we havef =xgand soCf =xCgand it sufﬁces to replacexbyx.

3) Assume thatf =xg for some x∈M (since, if xM, then xis invertible inT and soCgT =x⁻¹xT =T). Our aim is to show that CgT =T.

Sincex∈xT =Cf T =xCgR_M, then x=xa/s for some a∈Cg and s∈ R\M and so xa/s−1=0. Therefore, a/s−1∈Ann_Tx⊆M (since T M local) and thena/sM since 1M. This means thata/sis invertible inT and so CgT =T sincea/s∈CgT.

Proof of Theorem2.1. Notice that in T, principal and locally principal are the same concept (since T is local). Let f be a Gaussian polynomial of RX and assume either (i) T satisﬁes the property “every regular Gaussian polynomial has principal content ideal” andf is regular inRX, or (ii)D is a Prüfer domain and T satisﬁes the property “every Gaussian polynomial has principal content ideal”.

By Lemma 2.2, letx∈T and g∈RXsuch thatf =xg andCgT =T. Hence, it suffices to show that Cg is locally principal inR since Cf =xCg. Therefore, it suffices to show that Cg is R-projective, that is Cg⊗RT is T-projective and Cg⊗RR/MisR/M-projective (Glaz, 1989, Theorem 5.1.1(1)). SinceT isR-flat, thenCg⊗RT=CgT =T isT-projective. On the other hand,Cg⊗RR/M= Cg/MCg=Cg/MTCg=Cg/MT =Cg/M which is a finitely generated ideal of the domain R/M. It remains to show that Cg⊗RR/M=Cg/M) is R/M-projective in both cases (i) and (ii).

(i) IfxM, theng=f by the proof of Lemma 2.2(2) and sogis a Gaussian polynomial ofRX. If x∈M ⊆R, thenx is a regular element ofR (since f =xg andf is regular in RX) and so g is a Gaussian polynomial ofRX. Therefore, g is a Gaussian polynomial ofRXin all cases.

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Hence the image g¯ of g in R/MX is a Gaussian polynomial of R/MX by Heinzer and Huneke (1997, Remark 1.2) and thenC_R/Mg¯ =C_Rg/M=C_R⊗R

R/MisR/M-projective by Loper and Roitman (2004, Theorem 4) sinceD =R/M is a domain. Therefore,Cf is locally principal.

(ii) In this caseD =R/M is a Prüfer domain. It follows thatCg⊗R/M= Cg/M is R/M-projective since Cg/M is a ﬁnitely generated ideal of R/M.

Therefore,Cf is locally principal and this completes the proof of Theorem 2.1.

The hypothesis “qfD=T/M” cannot be omitted in Theorem 2.1(2) even if D is a ﬁeld (Example 2.3).

Let A be a ring,E be anA-module and R =A∝E be the set of pairsa e with pairwise addition and multiplication given by:a eb f =ab af+be.Ris called the trivial ring extension ofAbyE. Considerable work, part of it summarized in Glaz (1989) and Huckaba (1988) (where R is called the idealization of E byA), has been concerned with trivial ring extensions.

Example 2.3. Letkbe a proper subﬁeld of a ﬁeldK,T =K∝Kbe the trivial ring extension of K byK,M =0∝K, and letR =k∝K be the trivial ring extension ofkbyK. Then:

1) a) T is a local ring and M is the unique proper ideal ofT. In particular, every polynomial inT Xhas principal content ideal;

b) Every polynomial inT Xis Gaussian.

2) a)Ris a local ring with maximal ideal M;

b) Every polynomial inRXis Gaussian;

c) There existsf ∈RXsuch thatCf is not a principal ideal.

Proof. 1) a) T is a local ring by Huckaba (1988, Theorem 25.1(3)). Also, M=T01is the unique proper ideal of T by Huckaba (1988, Theorem 25.1(6)).

In particular, every polynomial in T Xhas principal content ideal.

b) Let f ∈T X. Our aim is to show that f is Gaussian. Let g∈T X. Two cases are then possible.

Case 1. Cf =T or Cg=T. In this case, Cfg=Cf Cg since every polynomial with principal content ideal is Gaussian.

Case 2. Cf =T and Cg=T. In this case, Cf ⊆M and Cg⊆M and soCfg=Cf Cg=0sinceM²=0.

Hence, Cfg=Cf Cg in all cases and so f is a Gaussian polynomial inT X.

2) a) By Huckaba (1988, Theorem 25.1(3)), R is a local ring with maximal idealM.

b) Argue as in the proof of 1)b).

c) Let a b∈K such that a b is a k-linearly independent set and set f = 0 a+0 bX∈RX. We claim that C_Rf =R0 a+R0 b=0∝ka+kb

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is not a principal ideal. Deny. Then, there existsc∈K such that ka+kb=kc. In fact, if there exists c∈K such that C_Rf =R0 c=0∝kc (since C_Rf ⊆M), thenka+kb=kc. Hence,ka+kb=kcand soa b is ak-linearly dependent set, a contradiction. Therefore,C_Rf is not a principal ideal and this completes the proof of Example 2.3.

Now we are able to construct a non-Noetherian local rings with zerodivisors verifying the property: every (regular) Gaussian polynomial has principal content ideal (cf. Examples 2.4 and 2.5). Recall that a ring is coherent if every ﬁnitely generated ideal of R is ﬁnitely presented. In particular, any Noetherian ring is coherent.

Example 2.4. Let D be a local domain which is not a ﬁeld, K=qfD, and let R =D∝K be the trivial ring extension ofD byK. Then:

1) Every regular Gaussian polynomial ofRXhas principal content ideal;

2) IfDis a valuation domain, then every Gaussian polynomial ofRXhas principal content ideal;

3) Ris not coherent. In particular,Ris not Noetherian;

4) Ris local. In particular,Ris not locally Noetherian.

Proof. LetT =K∝Kbe the trivial ring extension of KbyK. ThenT has M = 0∝K=R01 as a unique proper ideal. If h T →T/M K) is the canonical surjection, thenR =D∝K=h⁻¹Dand:

1) Follows from Theorem 2.1(1) since every ideal ofT is principal;

2) Follows from Theorem 2.1(2) sinceD is a valuation domain and every ideal of T is principal;

3) Ris not coherent by Kabbaj and Mahdou (2004, Theorem 2.8(1)). In particular, Ris not Noetherian;

4) Ris local by Huckaba (1988, Theorem 25.1) sinceD is local. In particular,R is not locally Noetherian and this completes the proof of Example 2.4.

Example 2.5. Let D be a non-Noetherian local domain which is not a ﬁeld, K=qfD,T =KX/Xⁿ=K+M be a local ring, where M=XT its maximal ideal andnbe a positive integer. SetR=D+M. Then:

1) Every regular Gaussian polynomial ofRXhas principal content ideal;

2) IfDis a valuation domain, then every Gaussian polynomial ofRXhas principal content ideal;

3) Ris local and not Noetherian. In particular,Ris not locally Noetherian.

Proof. 1) By Theorem 2.1(1) since every ideal ofT is principal.

2) By Theorem 2.1(2) sinceD is a valuation domain and every ideal ofT is principal.

3) Ris local sinceT andDare local. On the other hand,Ris not Noetherian since D is not Noetherian andR is a faithfully ﬂatD-module. Therefore, R is not locally Noetherian and this completes the proof of Example 2.4.

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ACKNOWLEDGMENT

We would like to thank the referee for his/her useful suggestions and comments, which have greatly improved this paper.

REFERENCES

Arnold, J. T., Gilmer, R. (1970). On the contents of polynomials. Proc. Amer. Math. Soc.

24:556–562.

Anderson, D. D., Kang, B. J. (1996). Content formulas for polynomials and power series and complete integral closure.J. Algebra181:82–94.

Cahen, P. J. (1988). Couple d’anneaux partageant un idéal.Arch. Math.51:505–514.

Corso, A., Heinzer, W., Huneke, C. (1998). A generalized Dedekind–Mertens lemma and its converse.Trans. Amer. Math. Soc.350:5095–5106.

Corso, A., Vasconcelos, W., Villarreal, R. (1998). Generic Gaussian ideals.J. Pure Appl. Alg.

125:117–127.

Fontana, M. (1980). Topologically deﬁned classes of commutative rings. Ann. Math. Pura Appl.123(4):331–355.

Gabelli, S., Houston, E. (1997). Coherentlike conditions in pullbacks. Michigan Math. J.

44:99–123.

Glaz, S. (1989).Commutative Coherent Rings. Lecture Notes in Mathematics. Vol. 1371.

Glaz, S., Vasconcelos, W. (1998). The content of Gaussian polynomials.J. Algebra202:1–9.

Heinzer, W., Huneke, C. (1997). Gaussian polynomials and content ideals.Proc. Amer. Math.

Soc.125:739–745.

Heinzer, W., Huneke, C. (1998). The Dedekind–Mertens lemma and the content of polynomials.Proc. Amer. Math. Soc.126:1305–1309.

Huckaba, J. A. (1988). Commutative Rings with Zero Divisors. New York-Basel: Marcel Dekker.

Kabbaj, S., Mahdou, N. (2004). Trivial extensions deﬁned by coherent-like conditions.

Comm. Algebra32(10):3937–3953.

Loper, K. A., Roitman, M. (2004). The content of a Gaussian polynomial is invertible.

Proc. Amer. Math. Soc.133(5):1267–1271.

Northcott, D. G. (1959). A generalization of a theorem on the content of polynomials.

Proc. Camb. Philos. Soc.55:282–288.

Rush, D. E. (2000). The Dedekind–Mertens lemma and the contents of polynomials.

Proc. Amer. Math. Soc.128:2879–2884.

Tsang, H. (1965). Gauss’s Lemma. Ph.D. thesis. Chicago, IL: University of Chicago.