High submodules and purity

A R K I V F O R M A T E M A T I K B a n d 7 n r 11

1.66 13 2 Communicated 26 October 1966 by T. ZqAGE~ and O. FROSTMA~

High submodules and purity

B y B o T. STErCSTROM

1. Introduction

The N-high subgroups of an abelian group G were defined by Irwin [5] as m a x i m a l subgroups having zero intersection with the given subgroup N of G. I n this note we extend some well-known relations between neat and N-high subgroups ([2], w 28 a n d [4], p. 327) to abelian categories and in particular to modules over general rings.

As an application we will generalize a characterization of intersections of neat sub- groups, due to l~angaswamy [7]. The t e r m "high" will here be used in a sense more general t h a n t h a t it has in [5].

Notation. ,'4 is an abehan category in which every object M has an injective envelope E(M). F o r a n y subobject L of M we consider E(L) as a well-defined subobject of E(M).

2. High suhobjects

Let M be an object in M with a given subobject K. A subobject L of M is called K-high if L n K = 0 and L is m a x i m a l with respect to this. K-high subobjects do exist for a n y K ([3], p. 360). We obviously have

Proposition 1. A subobject L o/ M is K-high i/and only i] the composed morphism K-->M-~M/L is an essential monomorphism.

Corollary. I / L is K-high in M, then (i) L + K is essential in M.

(ii) E(M)= E(L) 9 E(K).

The K-high subobjects of M m a y be described in terms of injective envelopes, as was done in [5] and [6] for abelian torsion groups.

Proposition 2. The K-high subobjects o/ M are just the intersections o / M with com- plementary summands o/ E(K) in E(M).

Proo/. I f L is K-high, then E(M)= E(L) 9 E(K) b y the corollary, and L = E(L) N M since also E(L) n M N K = 0 . Conversely, suppose E(M) = E ( K ) 9 H. Then H n M N K = 0 , a n d if L is K-high in M with L ~ H N M , then E ( L ) ~ E ( H N M ) = H . Clearly it follows t h a t E(L) = H , and H fi M = E(L) fi M =L is K-high.

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B. T. STE~STRfM, High submodules and purity

Proposition 3. Let L be a subobject o/ M. The/ollowing statements are equivalent:

(a) L is K-high/or some K in M .

(b) L is the intersection o / M with a direct summand o] E(M).

(c) L is essentially closed in M, i.e. L = E(L) N M.

(d) For every essential subobject H o / M such that L c H, also H/L--> M /L is essential.

Proo/. The proof of ( a ) ~ ( b ) ~ ( c ) is similar to the proof of prop. 2. (a)~(d): L e t H be an essential subobject of M with L c H . Suppose F is a n y subobject of M with F ~ L and F/L N H/L=O. This means t h a t F fi H = L , and we should show t h a t F = L . I f L is K-high in M, t h e n F f i K N H = L A K = 0 and hence F n K = 0 . B y the m a x i m a l i t y of L it follows t h a t F = L .

(d) ~(a): Choose a n y L-high subobject K of M. K + L is an essential subobject of M (eor. of prop. 1), so K + L / L - + M / L is essential b y hypothesis, a n d hence L is K-high (prop. 1).

3. High sequences

A short exact sequence O---~L---~M-+N---~O is called high if it makes L a K-high subobject of M for some K.

Proposition 4. The high sequences/orm a proper class.

Proo]. We have to verify the axioms for a proper class as given e.g. in [8]. Suppose ot :L--~M and fl : M-->N are monomorphisms. We will m a k e repeated use of prop. 3 (c) when verifying axioms P2 and 133, and of prop. 3 (d) when verifying 132 * and P3*.

P2. I f a and fl are high, t h e n fl~ is high. F o r L = E(L) N M = E(L) fi E(M) N 2V = E(L) N N.

P2*. I f flzr and M / L - ~ N / L are high, then fl is high. F o r if H is essential in N and i c H, then H/L-->N/L is essential and hence H / M ~ H / L / M / L - - ~ - N / L / M / L ~ N / M is essential.

P3. I f fi~ is high, then also ~ is. F o r if L = E(L) N 2V then L = E(L) N M.

P3*. I f fl is high, then also M/L--~N/L is. F o r if H/L is essential in N / L a n d H ~ M , then H is essential in M and hence H/L/M/L~=H/M-->N/M~2V/L/M/L is essential.

An object M is called simple if it has no subobjects except 0 and M. Following the terminology of [8] (sec. 9.6), we call a short exact sequence neat if every simple object is a relative projective for it.

Proposition 5. Every simple object is high-projective.

Proof. Suppose O-->L--~M--~P-~O is a high sequence w i t h P simple. If L is K-high in M, then K ~ P and K § = M , so the sequence splits.

Corollary. Every high sequence is neat.

As a result in the reverse direction we have

Proposition 6. Let 0 be a class of objects such that /or every M =~0 there exists a monomorphism P - ~ M /or some P # 0 in O. Then ~ - 1 ( 0 ) c {high sequences).

174

ARKIV FOR MATEMATIK. Bd 7 nr 11 Proo/. L e t O-->L-->M-->N->O be in

7/:--1(O).

E(L) N M/L-->O belongs to a - l ( O ) b y axiom P3 for proper classes. We want to use prop. 3 and therefore assume L 4=E(L) N M . B u t then there exists a m o n o m o r p h i s m qD :P-->E(L) N M / L w i t h P 4:0 in O. We m a y lift ~o to a monomorphism 95 :P-->E(L) N M . B u t t h e n I m ~ N L = 0, which is impossible.

Remark. The notion of high subobjects may, of course, be dualized. We call a sub- object L of M K-low if K § and L is minimal with respect to this. K-low subobjects do not necessarily exist for all K, unless A has projective envelopes.

B u t in a n y case one m a y verify t h a t the class of low sequences is a proper class, and t h a t every simple object is relatively injective for it.

4. High submodules

We now let A be the category of left modules over a ring A. Choosing O in prop. 6 to be the class of cyclic modules, we obtain

Proposition 7. Every pure sequence is high.

The t e r m " p u r e " is here used in the sense of [8] (see. 9.3). When A is commutative, we m a y take O to be Op = { A / I [ I prime ideal or (0)} and apply [1] (w 1), which gives Proposition 8. Let A be a noetherian, commutative ring. Then every Op-pure sequence is high.

Corollary. Let A be a noetherian, commutative ring where every prime ideal 4:0 is maximal. Then a short exact sequence is neat i/ and only i/ it is high.

5. Intersections o f neat submodules

I n this section A is assumed to be a noetherian, c o m m u t a t i v e ring with every prime ideal 4:0 maximal. So A is either a noetherian integral domain of Krull dimen- sion 1 or an artinian ring (assume not a field). As was found above, the concepts of neat and high sequences coincide over A, and this fact makes it possible to extend a result on intersections of neat subgroups of an abelian group, proved b y Rangas- w a m y [7], to modules over A.

L e t M be an A-module. Denote b y Soc M its socle and b y SocmM the homogenous component of Soc M determined b y the m a x i m a l ideal Ill. I f L is a n y submodule of M, E(L) r1 M is a minimal neat submodule of M containing L. We call it a neat hull of L in M. E v e r y neat submodule of M containing L also contains a neat hull of L in M. If N is a neat hull of L in M, then L contains the socle of N.

Proposition 9. Let L be a submodule o/ M and put O' = {A/m[ m maximal ideal with S o c m M ~ L ) . The/ollowing statements are equivalent:

(a) L is an intersection o/ neat submodules o/ M . (b) L is the intersection o/its neat hulls in M.

(c) M / L has no simple submodules o/type in 0'.

(d) L is O-pure in M (where O=O'U (0)).

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B. T. STE~STnhM, High submodules and purity

Proo/. T h e e q u i v a l e n c e of (a) a n d (b) is obvious. T h e c o n d i t i o n (c) m e a n s t h a t M / L is O-torsion-free, so t h e e q u i v a l e n c e of (c) a n d (d) follows f r o m [8] (prop. 6.2).

( a ) * ( c ) : S u p p o s e L = f~2V~ w i t h N~ n e a t in M . T h e n M / L = M / f ~ N i c I I M / N ~ , w h i c h is O - t o r s i o n - f r e e since e v e r y M/N~ is O - t o r s i o n - f r e e ([8], p r o p . 6.2).

(c) ~ (a): F o r e a c h x@L we h a v e t o f i n d a h i g h s u b m o d u l e of M c o n t a i n i n g L b u t n o t x. I f A x N L = 0 , t h i s is t r i v i a l t o do. So s u p p o s e A x NL 4 0 a n d p u t

I = ( a e A l a x e L } .

I is a p r o d u c t of p r i m a r y ideals, a n d e a c h p r i m a r y i d e a l c o n t a i n s a p o w e r of its r a d i c a l . T h e r e f o r e we h a v e n t l . . . m ~ x c L , w h e r e l~h a r e m a x i m a l ideals a n d l~2"...'l~x~=L. Choose a yell~2...m~x such t h a t y ~ L ; t h e n l u ~ y ~ L . I t will b e sufficient t o l o o k for a h i g h s u b m o d u l e c o n t a i n i n g L b u t n o t y. T h e h y p o t h e s i s (c) i m p l i e s t h a t A / m l ~ O ' , so t h e r e is a z ~ L w i t h l ~ h z = 0 . P u t u = y + z a n d let N b e a n e a t hull of L § in M . I f y ~ N , t h e n we a r e done. I f y e N , t h e n z e S o c N a n d h e n c e z e L § A u b y t h e r e m a r k j u s t p r e c e d i n g t h e p r o p o s i t i o n . W r i t e z = v § au w i t h v e L a n d n o t e t h a t a (F m l since o t h e r w i s e z = v + ay eL. W e g e t (1 - a) z = v § ay, w h i c h gives n h ( v + a y ) = 0 . Since v § a n d lUl is m a x i m a l , we conclude t h a t

L f i A ( v + a y ) = O . A n y A ( v § s u b m o d u l e N ' c o n t a i n i n g L will n o w do, b e c a u s e y r N ' .

Corollary. Let A be a noetherian integral domain o/ dimension 1. Suppose L is a torsion-/ree submodule o / M . L is then an intersection o/ neat submodules o / M i / a n d only i / A s s ( M / L ) = A s s ( M ) .

Proo/. A s s ( M ) d e n o t e s t h e set of p r i m e ideals which a r e a n n i h i l a t o r s of e l e m e n t s in M . W h e n L is torsion-free, 0 ' = ( A / m l m ~ A s s ( M ) } a n d t h e r e s u l t follows.

Dept. o] Mathematics, Royal Institute of Technology, Stockholm 70, Sweden

R E F E R E N C E S

1. BOURBAKI, N., Alg~bre commutative (ch. 4), Hermann, Paris 1961.

2. FvcHs, L., Abelian groups, Budapest 1959.

3. GABRIEL, P., Des categories ab@licnnes, Bull. Soc. Math. France, 90, 323-448 (1962).

4. H ~ I S O N , D. K., IRWin, J. M., PEE~C~, C. L., and WALKER, E. A., High extensions of abelian groups, Acta Math. Acad. Sci. Hung., 14, 319-330 (1963).

5. IRWIn, J. M., High subgroups of abelian torsion groups, Pac. J. Math., 11, 1375-1384 (1961).

6. I~wr~, J. M., and WALKER, E. A., On N-high subgroups of abelian groups, Pac. J. Math., 11, 1363-1374 (1961).

7. RA~GASWAMY, K. M., Characterization of intersections of neat subgroups of abelian groups, J. Indian Math. Soc., 29, 31-36 (1965).

8. STE~STR6M, B., Pure submodules, Arkiv fhr Matematik 7, 159 (1967).

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Tryckt den 25 maj 1967

Uppsala 1967. Almqvist & Wiksells Boktryckeri AB