Self-injective and PF endomorphism rings

In this paper, the rings which have a torsion theory T with associated torsion radical t such that R/t(R) has a minimal r-torsionfree cogenerator are studied. When T is the trivial torsion theory these are precisely the left QF-3 rings. For T = TL, the Lambek torsion theory, this class of rings is wider but, with an additional hypothesis on TL it is shown that if R has this property with respect to the Lambek torsion theory on both sides, then R is a (left and right) QF-3 ring. The results obtained are applied to get new characterizations of QF-3 rings with the ascending chain condition on left annihilators. A ring R is called left QF-3 if it has a minimal faithful left i?-module and left QF-3' if the injective envelope E(RR) is a torsionless module. These rings have been the object of extensive study (for example in [3, 9, 11, 12, 13, 16]) and, recently, Baccella has obtained in [2] structure results for the class of nonsingular, finite-dimensional QF-3 rings. In the present paper we consider a (hereditary) torsion theory r in i?-mod with associated torsion radical t and the property that R/t{R) has a minimal r-torsionfree cogenerator X, in the sense that X is a r-torsionfree module which cogenerates R/t(R) and is a direct summand of every r-torsionfree cogenerator of R/t(R). When r is the trivial torsion theory in which all i2-modules are r-torsionfree, this property defines left QF-3 rings. The class of rings which have this property with respect to the Lambek torsion theory TL is wider than the class of left QF-3 rings but when TL is strongly semiprime and R has this property for the Lambek torsion theory on both sides, This work was partially supported by the CAICYT (0784-84).

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“…https://doi.org/10.1017/S1446788700030718 [5] QF-3 rings and torsion theories 255 PROOF. Let X b e a minimal £^(fli?…”

Section: Proposition 4 Let R Be a Ring Such That Tl Is Strongly Semimentioning

confidence: 99%

QF-3 rings and torsion theories

Pardo

González

1989

J Aust Math Soc A

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“…We recall that a ring R is said to be left-Kasch (or a left S-ring) when each simple left R-module is isomorphic to a minimal left ideal. In [5,Theorem 3.1] it is shown that if M is a self-faithful 2-quasi-projective module, then 5 is left Kasch if and only if M is a finitely generated i?Z-module (7?Z-module means that M cogenerates each of its simple quotients). From this we get the following result.…”

Section: Re Is Not a Qf-3 Module Nevertheless If M Is Z-quasi-projementioning

confidence: 99%

“…Let P be a projective module and T its trace ideal on R. We recall that P is distinguished if, for xeP, Tx = 0 implies x=0 [7]. This is equivalent to P being self-faithful [5]. Thus we get the following partial improvement of [7,Proposition 6] and [7,Corollary 7].…”

Section: Re Is Not a Qf-3 Module Nevertheless If M Is Z-quasi-projementioning

confidence: 99%

“…Let M be a 2-quasi-projective module and T the smallest torsion class of o[M] containing all the modules of the form XIX M , with X in o [M], which in this case consists precisely of the modules N such that Hom R (M, N) = 0 [5]. A module N of o[M] is called M-faithful when it is T-torsion-free, that is, when Hom R (M, A")#0 for every non-zero submodule X of N. If M is M-faithful we will say that M is self-faithful.…”

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confidence: 99%

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QF-3 endomorphism rings of Σ-quasi-projective modules

Pardo

González

1988

Glasgow Math. J.

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A ring R is called left QF-3 if it has a minimal faithful left R-module. The endomorphism ring of such a module has been recently studied in [7], where conditions are given for it to be a left PF ring or a QF ring. The object of the present paper is to study, more generally, when the endomorphism ring of a 2-quasi-projective module over any ring R is left QF-3. Necessary and sufficient conditions for this to happen are given in Theorem 2. An useful concept in this investigation is that of a QF-3 module which has been introduced in [11]. If M is a finitely generated quasi-projective module and o [M] denotes the category of all modules isomorphic to submodules of modules generated by M, then we show that End( R M) is a left QF-3 ring if and only if the quotient module of M modulo its torsion submodule (in the torsion theory of o [M] canonically defined by M) is a QF-3 module (Corollary 4). Finally, we apply these results to the study of the endomorphism ring of a minimal faithful R-module over a left QF-3 ring, extending some of the results of [7].Throughout this paper R denotes an associative ring with identity, and /?-mod denotes the category of left R-modules. If M is a module, then we will say that a module N is M-generated (M-cogenerated) if it is a quotient (resp. a submodule) of a direct sum Af (/) (resp. direct product M 1 ) of copies of M. If N is M-cogenerated, then we will also say that N is M-torsionless and that M is a N-cogenerator. The full subcategory of /?-mod consisting of the submodules of M-generated modules will be denoted by o [M]; it is a locally finitely generated Grothendieck category [11]. We recall that a module N is M-projective (M-injective) if, for every quotient module (resp. submodule) X of M, the homomorphism Hom R (N, M)->Hom R (N, X) (resp. Hom R (M, N)-*Wom R {X, N)) is an epimorphism and, in particular, M is quasi-projective when it is M-projective. M is a projective object of a [M] precisely when it is Z-quasi-projective, that is, M (/) is quasi-projective for each set /. The largest M-generated submodule of a module X will be denoted by X M . E(N) will stand for an injective envelope of N in R-mod; if N belongs to o [M], then its injective envelope in this category is precisely E(N) M . A module is called finitely cogenerated (FC for short) if it has a finitely generated essential socle. When R R is injective and finitely cogenerated, R is said to be a left PF ring. The endomorphism ring of a module M will be denoted by S = End( R M) and we will use the convention of writing endomorphisms opposite scalars. We refer the reader to [2] and [6] for all the ring-theoretic notions used in the text.In [11], a module M is called a QF-3 module if there exists a minimal M-cogenerator t Work partially supported by the CAICYT (0784-84).Glasgow Math. J. 30 (1988) 215-220.https://www.cambridge.org/core/terms. https://doi

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“…Later on, various generalizations of the Morita theory were studied by many authors (see Fuller, 1974;García Hernández and Gó mez Pardo, 1987a;1987b;Wisbauer, 2000). If A is a locally finitely generated Grothendieck category, M 2 A and S ¼ End A (M ), then, without any assumption on M, it was proved in García and Saorin (1989) that Hom A (M, À ) induces an equivalence between certain quotient categories of A and Mod-S respectively.…”

Section: Introductionmentioning

confidence: 99%

Equivalences Induced by Adjoint Functors

Modoi

2003

Communications in Algebra

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Let A and B be two Grothendieck categories, R : A ! B, L : B ! A a pair of adjoint functors, S 2 B a generator, and U ¼ L(S ). U defines a hereditary torsion class in A, which is carried by L, under suitable hypotheses, into a hereditary torsion class in B. We investigate necessary and sufficient conditions which assure that the functors R and L induce equivalences between the quotient categories of A and B modulo these torsion classes. Applications to generalized module categories, rings with local units and group graded rings are also given here.

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Self-injective and PF endomorphism rings

Cited by 18 publications

References 15 publications

QF-3 rings and torsion theories

QF-3 rings and torsion theories

QF-3 endomorphism rings of Σ-quasi-projective modules

Equivalences Induced by Adjoint Functors

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