NEW ZEALAND JOURNAL OF MATHEMATICS Volume 32 (2003), 105-115

INJECTIVE A N D PROJECTIVE M ODULES RELATIVE TO A TORSION THEORY

P a u l E . B l a n d a n d P a t r i c k F . S m it h

(Received April 2002)

Abstract. Injective and projective modules are studied relative to a hereditary torsion theory r on Mod-7?. It is shown that every r-projective module is projective if and only if the torsion ideal of R is a semisimple direct summand of R. Hereditary torsion theories are determined for which every r-injective module is injective and several conditions are given that characterize right hereditary rings. We also show that the right global r-projective dimension of R is equal to the right global projective dimension of R modulo its torsion ideal.

The purpose of this paper is to investigate injective and projective modules relative to a hereditary torsion theory r on Mod-i?. In Section 1 we study conditions under which r-projective modules are projective while Section 2 is devoted to studying similar conditions for r-injective modules. In Section 3 we look at right hereditary rings in a torsion theoretical setting and in Section 4 the r-projective dimension of a module and the right global r-projective dimension of a ring are considered.

Throughout, R will denote an associative ring with identity 1 and i?-module or module will mean unitary right .R-module. I f r = (T,F) is a hereditary torsion theory on Mod-i?, then T t = {/ | I is a right ideal of R and R / I G T} will denote the (Gabriel) filter [6] of right ideals of R uniquely determined by r. Modules in T will be called r-torsion and those in F are said to be r-torsion free. An element m of a module M is a r-torsion element of M if there is an / G T t such that m l — 0. The set tT( M ) = { m G M \ mR is r-torsion} is a submodule of M and tT(R ) is an ideal of R called the r-torsion submodule and the r-torsion ideal, respectively. Moreover, M G T if and only if tT( M ) = M and M G F if and only if tT( M ) = 0. A submodule N of M is said to be r-dense in M if M / N is r-torsion and r-pure in M if M / N is r-torsion free. A class C of modules is closed under extensions if whenever 0 —> L —> M —> iV —>0is exact and L , N G C, then M G C. If r = (T,F) is a hereditary torsion theory on Mod-i?, then T is closed under submodules, homomorphic images, arbitrary direct sums and extensions while F is closed under submodules, arbitrary direct products, extensions and injective hulls.

In the following, r will denote an arbitrary hereditary torsion theory on Mod-i?, the category of unitary right i?-modules. Standard results on torsion theory can be found in [3], [7] and [15] while general information on rings and modules can be found in [1 ] and [1 2 ].

2000 A M S Mathematics Subject Classification: Primary 16D40, 16D50,16E60; Secondary 16D60. Key words and phrases: torsion theory, T-injective module, r —projective module, hereditary ring. This paper was written while the first author was on sabbatical leave visiting the University of Glasgow. He would like to thank the faculty and staff there for their kindness and hospitality.

106 PAUL E. BLAND AND PATRICK F. SMITH

1. Relative Projective Modules

An /2-module M is r-projective if for each exact sequence 0 —► L —> X N —> 0 of .R-modules, where L is r-torsion free, H om ^(M ,X ) —► Hom^(M, N ) —> 0 is exact. This definition of a r-projective module agrees with that given in [3], but such a module is called codivisible in [2]. Note that if M is a r-projective .R-module and N is a r-torsion submodule of M , then M / N is r-projective. A proof of this fact can be found in [2].

The following result from [13] will be used several times in what is to follow.

Theorem 1.1. An R-module M is r-projective if and only if M /M t T(R ) is a projective R /tT(R)-module.

The class of projective modules is closed under extensions. The following theorem shows that this is also true for the class of r-projective modules.

Theorem 1.2. If N is a submodule of M and N and M / N are r-projective, then M is t -projective.

Proof. The short exact sequence

0 -► ( N + M t T{R ) ) / M t T(R ) - » M / M t T{R ) -+ ( M / N ) / ( ( M / N ) t T(R ) ) -► 0

splits since ( M / N ) / ( ( M / N ) t T(R ) ) is, by Theorem 1.1, a projective R /tT(R )~ module. Now R /tr (R ) is a r-torsion free .R-module, so any projective .R/£T(.R)-module is a r-torsion free R-module. Using Theorem 1.1 again, we see that N / N t T(R ) is a pro­jective R /tT(R)-modu\e, so N /N t T(R ) is r-torsion free. But (N n M t T(R ) ) /N t r (R ) is a r-torsion submodule of N /N t T(R ), so N fl M t T(R ) = N tT(R ). Hence ( N + M t T( R ) ) / M t T(R ) ~ N / ( N n M t T(R ) ) = N / N t T(R ) and so M / M t T(R ) ~ N / N t T(R ) ® ( M / N ) / ( ( M / N ) t T(R ) ) . Therefore M / M t T(R ) is a projective R /tT(R)-mod\i\e, so in view of Theorem 1.1, M is a r-projective .R-module. □

We are interested in conditions under which the class of r-projective modules co­incides with the class of projective modules. We now give two equivalent conditions each of which is necessary and sufficient for this to hold.

Theorem 1.3. The following are equivalent for a torsion theory r on Mod -R .(a) Every r-projective R-module is projective.

(b) For every r-projective R-module M , tT( M ) is a semisimple direct summand of M .

(c) The ideal tr (R ) = eR for some idempotent e G R and tT(R ) is a semisimple R-module.

Proof. (a) = > (b) If M is a r-projective .R-module, then M / t T( M ) is r-projective and therefore projective. Hence, the exact sequence 0 —> tT( M ) —> M —> M / t T( M ) —> 0 splits. Consequently, tT( M ) is a direct summand of M and since M is projective, tT( M ) is also projective. If A/" is a submodule of tT(M ) , tT( M ) / N is r-projective and hence projective. Thus 0 —> N —> tT( M ) —> tT( M ) / N —> 0 splits, so tT( M ) is semisimple.

(b) = > (c) is clear, so let’s show (c) ==> (a). Suppose that (c) holds and let e be an idempotent element of R such that tT(R ) = eR. If M is a r-projective

TORSION THEORY 107

ii-module, then M / M t T(R ) is a projective i?/£r (i?)-module, so M / M t T(R ) is a direct summand of a free R /tT(R )~ module. It follows that M / M t T(R ) is a direct summand of a free .R-module, so M / M t T(R ) is a projective i?-module. Therefore, 0 —> M t T(R ) —> M —► M / M t T(R ) —> 0 splits and so M = M t T(R ) © N for some projective submodule N of M . Next, note that M t T(R ) is a homomorphic image of a direct sum of copies of tT(R ). Since tT(R ) is semisimple, there is a family {S *}^/ of simple and necessarily projective submodules of tT(R ) such that tT(R )This leads to M t T(R ) = (Bj^jUj where for each j G J there is an i E I such that Uj ~ Si. Hence, M t T(R ) is a projective R - module, so M is projective. □

The set of torsion theories on Mod-i? can be partially ordered. We write a < r if Ta C T r or equivalently if Fr C FCT. If tq denotes the torsion theory in which every module is considered to be torsion free and T\ is the torsion theory in which every module is viewed as being torsion, then tq < r < T\ for any torsion theory r on Mod-R . An interesting and often studied torsion theory on Mod-i? is the Goldie torsion theory [8 ]. This torsion theory is the smallest torsion theory tq on Mod-i? such that T Tg contains all the essential right ideals of R. A module is rc-torsion free if and only if Z ( M ) = 0 where Z ( M ) is the singular submodule [10] of M . Z (R r ) is an ideal of R and a r^-torsion free module is said to be nonsingular.

A torsion theory r is said to be faithful if tT(R ) = 0. For such a torsion theory r on Mod-i?, Theorem 1.1 shows that a module is r-projective if and only if it is projective. The following corollary to Theorem 1.3 indicates that the converse holds for the Goldie torsion theory.

Corollary 1.4. R is a right nonsingular ring if and only if every tq -projective R-module is projective.

Proof. If Z (R r ) = 0, then R is rc-torsion free, so the result follows immediately from Theorem 1.1. For the converse, suppose that every rc-projective i?-module is projective and let r 6 tTG (R ). Then, because of Theorem 1.3, there is an idempotent e e tTG(R ) such that rR = eR. I f I is an essential right ideal I of R such that el = 0, then I C (1 — e)R, so eR fl / = 0. Hence eR = 0 and, consequently, r = 0. Therefore Z (R r ) = 0. □

An i?-module M is said to be semi-artinian if S o c r (M /N ) ^ 0 for each proper submodule N of M and R is a right semi-artinian ring if R is semi-artinian as a right .R-module. One can prove that R is right semi-artinian if and only if every .R-module is semi-artinian [15, VIII, 2.5]. If T = { M \ M is semi-artinian} and F = { M \ Soc#(M ) = 0}, then ts = (T, F) is a torsion theory on Mod-i?. For this torsion theory, tTs (M ) is the largest and necessarily unique semi-artinian submodule of M . Theorem 1.3 takes a particularly nice form for semi-artinian torsion theories.

Corollary 1.5. The following are equivalent for the semi-artinian torsion theory ts on Mod-i?.

(a) Every Tg -projective R-module is projective.

(b) SoCtf(M) is a direct summand of M for every ts -projective R-module M .

(c) Socr (R ) = eR for some idempotent e 6 R.

108 PAUL E. BLAND AND PATRICK F. SMITH

Proof. (a) = > (b). I f every r^-projective .R-module is projective, Theorem 1.3 indicates that tTS( M ) is a semisimple direct summand of M . But So c r ( M ) C tTs(M ) , so it follows that Socr ( M ) — tT3(M ) .

(b) ==> (c) is clear, so let’s show (c) ==» (a). In light of Theorem 1.3, it suffices to show that Socr (R ) — tTs(R ). Now tTs(R ) is the unique, largest semi-artinian submodule of R and Socr (R ) is a semi-artinian submodule of R , so Socr (R ) C tTs(R ). Hence it must be the case that Socr (R ) = SocR(tTs(R ) ) . But any semi- artinian module is an essential extension of its socle [15, VIII, §2], so if Socr(R ) is a direct summand of R , then Socr (R ) = tTs(R ). □

A somewhat “stronger” condition on the class of r-projective modules is for it to coincide with the class of all -R-modules. Proposition 4.2.2 in [3] shows that every .R-module is r-projective if and only if R /tT(R ) is a semisimple ring. We conclude this section with an easily proved and similar result for right semi-artinian rings.

Theorem 1.6. Every right R-module is ts -projective if and only if R is right semi-artinian.

2. Relative Injective Modules

If R is a ring and r is a torsion theory on Mod-i?, then an .R-module M is said to be r-injective if for each short exact sequence 0 —> L —> X —> iV —> 0 of .R-modules, where N is r-torsion, Hom^(X, M ) —> Hornr ( L , M ) —> 0 is exact.

It was pointed out in Section 1 that factor modules of r-projective modules by r-torsion submodules are r-projective. A dual result holds for r-injective modules: r-pure submodules of r-injective modules are r-injective. This follows since if M is a r-pure submodule of a r-injective module E , then E t = { x e E \ x + M £ tT( E / M ) } is a r-injective submodule of E containing M , E T/ M is r-torsion and E / M is r-torsion free. Thus E T/ M C E / M shows that M = E T.

If E ( M ) is the injective envelope [5] of M and we set E — E ( M ) and E T = E r (M ) , then E T( M ) is the r-injective envelope [3], [7] of M . In this case, E T( M ) is a r-dense essential extension of M that is unique up to isomorphism.

We saw in Theorem 1.3 that every r-projective .R-module is projective if and only if tT(R ) is a semisimple direct summand of R. We now characterize the torsion theories for which every r-injective .R-module is injective.

Theorem 2.1. If tq is the Goldie torsion theory on Mod -R , then for any torsion theory t , every r-injective R-module is injective if and only if tq < r.

Proof. Let tq < r, and suppose that M is r-injective. If I is a right ideal of R and / : I —► M is an .R-linear mapping, let K be a complement of I in R. Then I © K is an essential right ideal of R and so is in T Tg C T t . Extend f to g : I ® K —> R by g(r + k) = f ( r ) , r £ 7, k € K, and then extend g to h : R —> M by using the r-injectivity of M . If h (l) = m, then / (r) = mr for all r G /, so Baer’s Criterion[10], [15] shows that M is an injective .R-module.

Conversely, suppose that every r-injective module is injective and let M be a r-torsion free .R-module. If / is an essential right ideal of R and i : I —► R and j : I —» E ( I ) are canonical injections, then there is an injective .R-linear mapping g : R —> E ( I ) such that g o i — j. I f R is identified with its image in E ( I ) , then we have R / I C E ( I ) / I . Since r-injectives are injective, E T( I ) = E ( I ) , so

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E ( I ) / I is r-torsion and therefore R / I is r-torsion. But M is r-torsion free, so Hom .n (R /I ,M ) = 0. Next, suppose that m is a nonzero element of M such that m l = 0. If / : R / I —■» M is such that r + I i— > mr, then 0 ^ f G Hornr (R / I , M ). Hence it must be the case that m l ^ 0 for all m G M , m ^ 0. Consequently, the singular submodule of M is zero, so M is r^-torsion free. Therefore the r-torsion free class of r is contained in the r^-torsion free class of t q , so t q < r.' □

Corollary 2.2. The following are equivalent for the semi-artinian torsion theory ts on Mod-it!.

(a) Every ts -injective R-module is injective.

(b) The R-module R / I has nonzero socle for every proper essential right ideal I of R.

Proof, (a) =>- (b) Since every r^-injective /^-module is injective, tq < ts and so T rs contains every essential right ideal of R. Thus if I is a proper essential right ideal of R , then R / I is semi-artinian and so Socr ( R / I ) ^ 0.

(b) = > (a) Suppose that S o c r (R / I ) ^ 0 for every proper essential right ideal I of R. I f I is an essential right ideal of R, suppose that R ' is a nonzero homomorphic image of R /I . Then there is a proper and necessarily essential right ideal J of R containing I such that R /J ~ R ' . Hence Socr (R ' ) ^ 0 which shows that R / I is a semi-artinian /^-module. Therefore I G T Ts and we have tq < Ts. Theorem 2.1 now indicates that every rg-injective module is injective. □

A torsion theory is said to split if the torsion submodule of each injective module is injective. Theorem 2.1 produces a condition sufficient for this always to happen.

Corollary 2.3. If t is a torsion theory on Mod-R such that t q < t , then r splits.

Proof. The r-torsion submodule of an injective module is r-injective and Theorem 2 . 1 indicates that this submodule is injective. □

It is not necessary for tq < r in order for the r-torsion submodule to split off from each injective module. This is illustrated by the following example.

Example 2.4. I f R is a commutative Noetherian ring and I is an ideal of R , Matlis has shown in [11] that the torsion submodule of each injective module is injective for the /-adic torsion theory [15] on Mod-R . If Z is the ring of integers and p is a prime in Z, consider the (p)-adic torsion theory r(p) on Mod-Z with filter T Tip) = {(j>fc) I k a non-negative integer}. In light of what is known for commutative Noetherian rings, if M is an injective Z-module with T(py-torsion submodule £r (M ), iT(p)(M ) is a direct summand of M . But every nonzero ideal of Z is essential, so the filter of ideals for the Goldie torsion theory tq on Mod-Z is not contained in the filter of ideals of r(p) .

A torsion theory r is cohereditary if every homomorphic image of each r-torsion free module is r-torsion free. For this type of torsion theory, the condition that every r-injective module is injective yields information about the ring.

Theorem 2.5. If r is a cohereditary torsion theory on Mod-R and every t-injective R-module is injective, then R /tT(R ) is a semisimple ring.

110 PAUL E. BLAND AND PATRICK F. SMITH

Proof. Let r be cohereditary and suppose that every r-injective .R-module is in­jective. I f M is r-torsion free, then E T( M ) is also r-torsion free, so that E T( M ) / M is both r-torsion and r-torsion free. Therefore M = E T(M ) , so we see that every r-torsion free jR-module is r-injective and hence injective. I f I / t T(R ) is a right ideal of R /tT(R ), then I / t T(R ) is a r-torsion free .R-module and so I / t T(R ) is an injective .R-module. It follows that I / t T(R ) is an R /tT(R ) direct summand of R /tT(R ). □

Corollary 2.6. If r is a cohereditary torsion theory on Mod-R and every r-injective R-module is injective, then every R-module is r-projective.

It was pointed out in Section 1 that the class of r-projective modules is closed under extensions. Since this is the case for injective modules, one wonders if the class of r-injective modules is closed under extensions. The following theorem shows that this is indeed the case.

Theorem 2.7. If N is a submodule of an R-module M such that N and M / N are both r-injective, then M is r-injective.

Proof. Suppose that M is a T-dense submodule of X. Then the sequence0 —» M / N —> X / N —> X / M —> 0 splits since M / N is r — injective. Hence there is a submodule Y of X containing N such that X / N = ( M / N ) ® (Y / N ) and Y / N ~ X /M . Thus N is a r-dense submodule of Y and, since N is r-injective, the sequence 0 —> N —* Y —» Y / N —>0 splits. Hence there is a submodule Z of Y such that Y — iV ® Z. Now X / N = ( M / N ) © ( Y / N ) implies that X — M + Y and M H Y = N , so it follows that X — M © Z. This shows that when N and M / N are r-injective, M is a direct summand of any module in which it is a r-dense submodule. But M is a r-dense essential submodule of E T( M ), so M = E T(M ) . Therefore M is r-injective. □

The Generalized Baer’s Criterion [3], [7] for r-injective modules indicates that an .R-module M is r-injective if and only if for each I <E T t and every R -linear map / : I —> M there is an m € M such that f ( r ) = mr for all r € I. If ts is the semi-artinian torsion theory on Mod-i?, it follows that T Ts contains every maximal right ideal of R. We now show that when investigating the ts—injectivity of a module, only R -linear mappings defined on maximal right ideals of R need be considered.

Theorem 2.8. An R-module M is ts -injective if and only if for each maximal right ideal I of R and every R-homomorphism f : I —* M there is an m E M such that f ( r ) — mr for all r € I .

Proof. If M is a T5 -injective .R-module, the stated condition follows from the Generalized Baer’s Condition. Conversely, suppose that the stated condition holds for an .R-module M and let I € T Ts. Suppose also that / : I —> M is an R -linear mapping. Next let S be the non-empty set of all ordered pairs (K ,g ) where K is a right ideal of R containing I, g : K ^ M is an .R-linear mapping and g restricted to / gives /. Partially order S by (K\,gi ) < (K 2,g2) if and only if K i C K 2 and g-2 restricted to K\ produces g\. Then Zorn’s Lemma can be used to show that S has a maximal element, say (K ,g ). If K ± R, then since I G T Ts and / C K, it follows that K € T Ts- Thus there is a right ideal J of R containing K

TORSION THEORY 111

such that J /K is a simple it!-module. If x + K is a nonzero element of J /K , then R / (K : x) ~ J /K and ( K : x) — {r G R \ xr G K } is a maximal right ideal of R. Hence if we let g : ( K : x) —> M be defined by g (r ) = g(xr ), then there is an m G M such that g(r ) = mr for all r G ( K : x). Finally, define <p : ( K + xR ) —> M by ip(k + xr) = g(k) + mr and note that ip extends g to K + xR. But this contradicts the maximality of (K ,g ) and so it must be the case that K = R. Invoking the Generalized Baer’s Condition again gives the result. □

3. Right Hereditary Rings and Torsion Theory

A ring R is right hereditary if every right ideal of R is projective. Cartan and Eilenberg have shown in [4] that such rings are characterized (1) by the fact that submodules of projective modules are projective and ( 2 ) by the fact that factor modules of injective modules are injective. When R is a right hereditary ring, r-projective and r-injective modules also enjoy these properties.

Theorem 3.1. If R is a right hereditary ring, then the following hold for any torsion theory r on Mod-R.(a) Every submodule of a r-projective module is r-projective.

(b) Every homomorphic image of a r-injective module is r-injective.

Proof, (a) Let M be a r-projective i?-module. Then there is a free -R-module F and a r-torsion submodule N of F such that M is a direct summand of F / N [13]. Consequently, if L is any submodule of M , then there is a submodule X of F containing N such that L ~ X /N . Since R is hereditary, X is projective and so since N is r-torsion, L is r-projective.

(b) Suppose that M is a r-injective ii!-module and let iV be a submodule of M . I f E ( M ) is the injective envelope of M, then since R is right hereditary, E ( M ) / N is an injective /^-module. But an /2-module X is r-injective if and only if E ( X ) / X is r-torsion free, so E ( M ) / M is r-torsion free. Hence, M / N C E ( M / N ) C E ( M ) / N gives E { M / N ) / { M / N ) C ( E { M ) / N ) / ( M / N ) « E ( M ) / M and so E ( M / N ) / ( M / N ) is r-torsion free. Thus M / N is r-injective. □

Kaplansky has shown in [9] that for a right hereditary ring, submodules of free .R-modules are isomorphic to a direct sum of right ideals of R. This theorem carries over to a torsion theoretical setting in a straightforward manner.

Theorem 3.2. If R is a ring in which every right ideal is r-projective, then ev­ery r-torsion free submodule of a free R-module is isomorphic to a direct sum of ( necessarily r-torsion free) right ideals of R.

Proof. The proof of Proposition 4.17 given in [14] for hereditary rings works with only minor changes required to accommodate the torsion theory. □

For a torsion theory r on Mod-i?, we call a ring R right r-hereditary if every r-dense right ideal of R is projective. Of course every right hereditary ring is right r-hereditary for any torsion theory r on Mod-i? and any ring R is right ro-hereditary. A ring R is also right r-hereditary for any torsion theory r on Mod—R for which every cyclic r -torsion i?-module is projective.

Right hereditary rings are closely linked to the Goldie torsion theory. Indeed, a ring R is right hereditary if and only if it is right r-hereditary for any torsion

112 PAUL E. BLAND AND PATRICK F. SMITH

theory r on Mod i? such that tq < r. The necessity of this is obvious, so for the converse suppose that R is right r-hereditary where r is a torsion theory on Mod-i? such that tq < t . If 7 is a right ideal of R and J is a complement in R of 7, then 7 © J is an essential right ideal of R. Hence, 7 © J E T tq C T t and so 7 © J is projective. Thus, 7 is projective, so R is right hereditary. The proof of the following theorem concerning right r-hereditary rings is an easy adaptation of the proof of Proposition 4.23 given in [14] for hereditary rings.

Theorem 3.3. The following are equivalent for a torsion theory r on Mod-i?.

(a) R is right r-hereditary.

(b) r-dense submodules of r-projective R-modules are t - projective.

(c) Factor modules of r-injective R - modules by t - torsion free submodules are t -injective.

Since hereditary rings are precisely those rings in which the tq -dense right ideals of R are projective, one wonders what class of rings is characterized by the “weaker” condition that the tg-dense right ideals of the ring are only T£-projective? Rings for which r-torsion free submodules of r-projective .R-modules are r-projective are also connected to right hereditary rings.

Theorem 3.4. If t is a torsion theory on Mod -R , then R /tT(R ) is a right hered­itary ring if and only if r-torsion free submodules of r-projective R-modules are r-projective.

Proo f. Suppose that R /tT(R ) is right hereditary and let M be a r-projective .R-module. Then Theorem 1.1 shows that M / M t T(R ) is projective as an R /tT( R ) - module. If N is a r-torsion free submodule of M , then(AT + M t T( R ) ) /M t T(R ) is a projective R /tT(R )~module. Now N fl M t T(R ) = 0, so it follows that ( N + M t T(R ) ) / M t T(R ) ~ N / ( N fl M t T(R )) ~ N . Hence N is a projective i?/tr (.R)-inodule. But N tT(R ) = 0 and so it follows from Theorem 1.1 that N is a r-projective /2-module.

Conversely, let I / t T(R ) be a right ideal of R /tT(R ). Then I / t T(R ) is a r-torsion free submodule of the r-projective .R-module R /tT(R ). Now [ I / t T(R ) ) t T(R ) = 0, so Theorem 1.1 shows that I / t T(R ) is a projective R /tT(R ) -module. □

One immediate consequence of Theorem 3.4 is that if R /tr (R ) is a right heredi­tary ring, then every r-torsion free submodule of a free right i? -module is r-projective and this in turn implies that every r-torsion free right ideal of R is r-projective. However, neither of these conditions need imply that R /tT(R ) is right hereditary. The following example shows that torsion theories exist for which r-torsion free submodules of free .R-modules are r-projective and yet R /tT(R ) is not a right hereditary ring.

Exam ple 3.5. Let D be a commutative domain that is not a Dedekind domain, suppose that Q is the field of fractions of D and consider the matrix ring R = { ( g £ ) | a € T ) and 6 , c € Q }. I f rs is the semi-artinian torsion theory on Mod-72, then tTs(R ) consists of all matrices in R with a = 0. Consequently, R /tTs(R ) is isomorphic to D and so R /tTs(R ) is not a right hereditary ring. How­ever, tTs(R ) is an essential right ideal of R and it follows from this that if F is a

TORSION THEORY 113

free -R-module, then tTs( F ) is an essential submodule of F. Hence, 0 is the only Ts—torsion free submodule of F and so every 7 5 -torsion free submodule of F is ts-projective (and in fact projective.) Note that this example also shows that even if r-torsion free submodules of free .R-modules are projective, R /tT(R ) may not be right hereditary.

4. Relative Projective Dimension

Let r be a torsion theory on Mod-P. If 0 —> K — >0is exact with

X projective, then 0 —► K /tT{ K ) —» X / tT( K ) M —> 0 is exact with X / tT(K ) r-projective and ker r-torsion free. This observation allows us to construct an

exact complex • • • —> P 2 -+ Pi Po M —> 0 in which Pn is r-projective and ker Sn is r-torsion free for each integer n > 0. We refer to such a complex as a r-projective resolution of M and say that a r-projective resolution of M is finite and of length n if there is a non-negative integer n such that Pk — 0 for all k > n and Pfc 7 ̂0 for 0 < k < n. I f M has at least one finite r-projective resolution, then the r-projective dimension of M , written r-pd/i(M ), is defined to be the length n of the shortest such resolution of M . I f no finite r-projective resolutions of M exist, then r-pd ft(M ) = 0 0 . The right global r-projective dimension of R is defined by rgl r -p d n (R ) = sup {r-pd^ (M ) | M is an R -module}. I f the projective dimension of an /2-module M and the right global dimension of R are denoted by pd^ (M ) and rgl pdji(R), respectively, then r i-p d ^ (M ) = 0 and ro-pd^(M ) = pd^ (M ) for every jR-module M . Hence, rgl r i-p d fl(P ) = 0 and rgl To-pdji(R) = rgl pd^(/?).

Lemma 4.1. If r is a torsion theory on Mod - R and

------> P 2 ^ P i ^ P 0 h M —► 0 (1)

is a r-projective resolution of M , then M / M t T(R ) has an R /tT(R)-projective res­olution of the form

------>P2 ^ P l ^ Po/Potr{K) 5 M /M t r (R ) -> 0 (2)where (a: + PotT(R )) = 60(2:) + M / M t T(R ) for each x + Potr (R ) G Po/PotT(R ).

Proof. From the construction of (1), we have that 0 —■> Kq —> Po ^ M —> 0 is exact where Kq is the kernel of <50- It follows that 0 —> (Kq + P0tT(R ) ) / P 0tr (R ) —■»

Po/PotT(R ) ^ M / M t T(R ) —■> 0 is exact. But K q is r-torsion free and PotT(R ) is r-torsion, so (K q + PotT(R ) ) / PotT(R ) ~ K q. Hence, we have an exact sequence

0 —> Kq —» Po/PoiT(P ) M / M t T(R ) —> 0 and Theorem 1.1 shows that Pq/PotT(R ) is projective as an P/tT(P)-module. It also follows from the technique used in the construction of (1) that 0 —■> Ki —» Pi —> Kq —> 0 is exact where Kq and K\ are r-torsion free and K\ is the kernel of <5i. Hence the induced sequence 0 —> (K\ + PitT(R ) ) / P i t T(R ) —► Pi/P\tT(R ) —> K q /K 0tT(R ) —* 0 is exact. But the class of r-torsion free modules is closed under extensions [3], so P\ is r-torsion free. Therefore KotT(R ) = P\tT(R ) = 0, so 0 —» K\ —> P i —> Kq 0 is an exact se­quence of R /tT(P)-modules and P i is a projective R /tT(R)-modu\e. Consequently,

we have an exact sequence 0 —> K\ —>■ Pi Pq/ P otT(R ) ^ M / M t T(R ) —► 0 where

114 PAUL E. BLAND AND PATRICK F. SMITH

P i and P0/ P 0tT(R ) are projective R /tT(R )~ modules. The existence of (2) now follows by induction. □

Theorem 4.2. If t is any torsion theory on Mod-R, then

T - p d R ( M ) = pd R/tT(R ) (M / M t T(R ) )

for any R-module M .

Proo f. If M is an .R-module such that r -p d R( M ) = n, let 0 —>• Pn ^ Pn- i —>

• • • —► Pq —* KI 0 be a r-projective resolution of M . Then Lemma 4.1 indicates

that we have an R /tT(R )~projective resolution of M / M t T(R ) of the form 0 —> Pn ^

Pn_ i P0/P 0tT(R ) S M / M t T(R ) -> 0. Hence, pdR/tT(R) ( M / M t T(R ) ) <r—pdfl(M ).

Conversely, suppose that pdR/tT^ [ M / M t T(R ) ) = n and let

o - Qn ^ Qn-1 - -------> Qo ™ M / M t r(R) 0 (3 )

be an .R/tT (.R)-projective resolution of M / M t T(R ). Next, construct an exact se­

quence 0 —> K n ^ Pn_i —> • • • —► P0 ^ M —> 0 of /^-modules where Pk is r-projective for k = 0,1, . . . , n — 1 and K n is r-torsion free. Then arguments similar to those given in the proof of Lemma 4.1 can be used to obtain an exact sequence

0 Kn h Pn-1 -> -----> P0/P0tr(R ) 5 M/MtT(R) 0 (4)

where Pq/PotT(R ) and P k, for k = 1, . . . , n — 1, are projective R /tT(R )~ modules. An application of the long version of Schanuel’s Lemma [12, 7.1.2] to (3) and (4) shows that K n is a projective R /tT(R)~module. Since Theorem 1.1 indicates that K n is a r-projective .R-module, we have T-pdR( M ) < pd R/tr(R ) (M /M t T{R ) ) . □

Theorem 4.2 has several consequences, the proofs of which are left to the inter­ested reader.

Corollary 4.3. If r is any torsion theory on Mod -R , then(a) rgl r -p d R(R ) = rgl pdR/tr(R)(R / tT( R ) ) ,

(b) R /tT(R ) is a right hereditary ring if and only if rgl T--pdR(R ) < 1, and

(c) rgl r-pdft(.R) = sup {r-pd# (M ) | M is a cyclic R-module } = sup{r-pdR(R /1 )I is a right ideal of R such that I D tT(R ) } .

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TORSION THEORY 115

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Paul E. Bland ^ „t-, . T7- i i tt ■ Patrick F. SmithEastern Kentucky University TT . . .Richmond University of GlasgowKentucky <?lasgow^ g Scotland

, , , , . pfs@maths.gla.ac.uk paul. bland<9eku .edu. , , , . http://www.gla.ac.ukhttp://www.eku.edu ' °