A structure theorem for left modules over left hereditary left perfect right coherent rings

We characterize the left Noetherian rings by the existence of decompositions of left modules into direct sums of an injective submodule and a submodule containing no injective submodule except 0. We also prove that a left Noetherian ring is left hereditary iff the suspension of each left ideal (see [7]) is injective or, equivalently, the above decomposition is unique.0. Let A be an Abelian group.Then A = D B, where D is divisible and B is reduced in the sense of Kaplansky [8]; that is, B has no divisible submodule except 0. If this holds for left modules over a unitary ring R, with "injective" and "/-reduced" replacing "divisible" and "reduced", respectively, then R must be left Noetherian.The proof requires the theorem of Eilenberg-Bass (see, e.g., Theorem 20.1, p. Ill, of [2]) which characterizes left Noetherian by the requirement that direct sums of injective left modules are injective.Furthermore, if the decomposition, which will be referred to as an i-decomposition, is unique in the sense that D is the unique injective submodule of A for which A = D ffi B, with B /-reduced, then every sum of injective submodules is injective, and hence R must be left hereditary by a theorem of Matlis [9].At the same time, it will be shown that a left Noetherian ring is left hereditary iff the suspension of each left ideal is injective.

Throughout this note, R stands for a ring with identity, and all modules are unital R-modules. In this note, we ask when the class of all finitely generated reflexive (torsionless) left modules is closed under extensions. This is not always the case, even if R is left and right noetherian (cf. Bass [2, Lemma 3.2] and [3, p. 12]). As will be shown, in case R is a commutat ive noetherian local domain of Krull dimension two, R is a CohenMacaulay ring whenever the class of the finitely generated reflexive modules is closed under extensions. Our main aim is to show that if R is a left and right noetherian ring then both the class of all finitely generated torsionless left modules and the class of all finitely generated reflexive left modules are closed under extensions if and only if for i = 1, 2 the functor Ext , 1 (Ext~ ( , R), R) vanishes on the finitely generated right modules. Note that the latter condition is satisfied if R is a commutat ive Gorenstein ring (see Bass [3]). We will show also that if R is a left and right noetherian ring with a minimal injective resolution 0 ~ RR ~ Eo ~ E1 , . . . such that weak dim E l < i + 1 for i = 0, 1 then both the class of finitely generated torsionless left modules and the class of finitely generated reflexive left modules are closed under extensions. In the proof, it will be shown that if R is a left and right noetherian ring with inj dim RR _--< 2 then the class of the finitely generated reflexive left modules is closed under extensions. On the other hand, even if R is a left and right noetherian ring with inj dim RR = inj dim R R < 2, the class of all finitely generated torsionless left modules is not necessarily closed under extensions. In what follows, we denote by ( )* both the R-dual functors, and for a given module M we denote by eu: M ~ M** the usual evaluation map and by E(M) the injective envelope of M. Recall that a module M is said to be torsionless if eu is a monomorph i sm and to be reflexive if eu is an isomorphism. Note that a module M is torsionless if and only if it is cogenerated by R. We denote by mod R (resp. mod R ~ the category of the finitely presented left (resp. right) modules. Note that if R is left noetherian then every finitely generated left module is finitely presented.

A left module M over a ring R is cofaithful in case there is an embedding of R into a finite product of copies of M. Our main result states that a semiprime ring R is left Goldie, that is, has a semisimple Artinian left quotient ring, if and only if R satisfies (i) every faithful left ideal is cofaithful and (ii) every nonzero left ideal contains a nonzero uniform left ideal. The proof is elementary and does not make use of the Goldie and LesieurCroisot theorems. We show that (i) and (ii) are Morita invariant. Moreover, (ii) is invariant under polynomial extensions, and so is (i) for commutative rings. Absolutely torsionfree rings are studied. The ring Q is a left classical quotient ring for the ring R C Q if every regular element (nondivisor of zero) of R is invertible in Q and if every element of Q is of the form b ~ιa where a,b E R and b is regular; in this case we also say that R is a left order in Q. A ring is said to be left Goldie if it has the ascending chain condition on left annihilators and has finite uniform dimension. (A left R -module has finite uniform dimension if it has no infinite direct sum of nonzero submodules, and it is said to be uniform if it is nonzero and any two nonzero submodules have a nontrivial intersection.) A theorem of Goldie [8,9] and Lesieur and Croisot [12] states that a ring is a left order in a semisimple Artinian ring if and only if it is semiprime and left Goldie. It is known that the ascending chain condition on left annihilators is not preserved under an equivalence of categories (Morita invariant); in fact, it does not go up to matrix rings. It is unknown whether being left Goldie is Morita invariant. In section two we give a proof of the theorem stated in the abstract, and in the prime case we give a proof which shows directly that such a ring is an order in a full matrix ring over a division ring. We also weaken the hypothesis of an important theorem on semiprime PI rings. In the third section we use these techniques to study absolutely torsion-free rings. In particular, we show that an absolutely torsionfree ring is Goldie if and only if it has a uniform left ideal, and that the endomorphism ring of a finitely generated projective module over an absolutely torsion-free ring is absolutely torsion-free.

Recall that a left module A (resp. right module B) is said to be singly injective (resp. singly flat) if $$Ext^{1}_{R}(F/K, A) = 0$$ (resp. $${\text {Tor}}_{1}^{R}(B, F/K)= 0$$ ) for any cyclic submodule K of any finitely generated free left R-module F. In this paper, we continue to study and investigate the homological objects related to singly flat and singly injective modules and module homomorphisms. Along the way, the right orthogonal class of singly flat right modules and the left orthogonal class of singly injective left modules are introduced and studied. These concepts are used to extend the some known results and to characterize pseudo-coherent rings and left singly injective rings. In terms of some derived functors, some homological dimensions are investigated. As applications, some new characterizations of von Neumann regular rings and left PP rings are given. Finally, we study the singly flatness and singly injectivity of homomorphism modules over a commutative ring.

A left QF-3 ring R is one in which RR, the ring considered as a left module over itself, can be embedded in a projective infective left R module Q(RR). QF-3 rings were introduced by Thrall [14] and have been studied and characterized by a number of authors [5, 8, 9, 12, 13, 15] usually restricted to the case of algebras over a field. In such a case, the concept of left QF-3 and right QF-3 coincide.The study of QF-3 rings and algebras and many other such classes of rings had its origin in the now classic papers of Nakayama [10, 11]. He was an outstanding pioneer in algebra for many years, and we acknowledge our great debt to him and to his many excellent papers.

We show that if R is a left Noetherian ring, then Here, for a Noetherian left module A, len A is its ordinal valued length as defined by Gulliksen (Gulliksen, (1973). T. H. A theory of length for noetherian modules. J. Pure Appl. Algebra 3:159–170), and ⊗ is the natural product on ordinal numbers.

Let R be a left and right artinian ring with identity, and / the Jacobson radical of R. In [4], M. Harada has considered a left serial ring R satisfying a condition (*, 2) that every maximal submodule of a direct sum of any two hollow modules is also a direct sum of hollow modules, and characterized such a ring by the structure of eR for each primitive idempotent e. Further it has been shown that the condition (*, 2) is equivalent to saying that every factor module of ejQ)eR is a direct sum of hollow modules for every primitive idempotent e. Modifying this, we here consider the following condition on a projective indecomposable right module eR over a ring R.

A left artinian ring Λ is projectively stable if no non-zero morphism f: M → N of finitely generated left Λ-modules M,N having no non-zero projective direct summands factors through a projective module. Such rings have many good properties and, although the definition is given in terms of the category of modules, the structure of projectively stable left artinian rings allows a satisfying description: they are “built” from well-known left artinian rings, left hereditary and serial, by a pull back of rings construction. We show that representations of projectively stable artin algebras are also “built” from representations of well-studied artin algebras, hereditary and serial. Namely, we give a simple geometric construction that produces the Auslander-Reiten quiver or the ordinary quiver of a projectively stable algebra from those of an hereditary algebra and a serial algebra.

An associative ring R with identity is said to be a left FTF ring when the class of the submodules of flat left R-modules is closed under injective hulls and direct products .We prove (Theorem 3.5) that a strongly graded ring R by a locally finite group G is left FTF if and only if Re is left FTF, where e is the neutral element of G.This provides new examples of left FTF rings.Some consequences of this Theorem are given. . IntroductionIn the papers [5], [6] we started the study of the rings R with the property that the class of the submodules of flat left modules, .PO, is closed under direct products and injective hulls.For these rings, .Po is the torsionfree class for some hereditary torsion theory on R-Mod.Thus we can use techniques of localization relative to this torsion theory to investigate these rings.This class of rings is large and we call them left FTF rings.Clearly, QF and regular rings are left and right FTF rings .In the commutative case, Enochs showed [3] that a noetherian commutative ring R is FTF if and only if Rp is Gorestein for every minimal prime ideal P of R. Noncommutative examples of left FTF rings can be found in [5], [6] .The aim of this note is to construct new examples of left FTF rings .The tool will be the strongly graded ring theory.The paper is organized as follows .In Section 1 we record the fundamental results on left FTF rings showed in [5] and [6] which we will use in the rest of the paper.We recall also the basic machinery from torsion theories and graded rings that we will use .The main result, Theorem 2.5, appear in Section 2 .Finally, we give some applications of this result in Section 3.

In this paper, we study the strong regularity of left SF rings and obtain the following results: Let R be a left SF ring. If R satisfies one of the following conditions, then R is a strongly regular ring: 1) R is a left WPZI ring; 2) R is a right WPZI ring; 3) R is a right weakly semicommutative ring; 4) R is a semicommutative ring; 5) R is a reversible ring.

We study an $R$-module $M$ such that for every proper submodule $K$ of $M$, Hom$_R(M/K,M)\neq 0$. Such modules are called coretractable. Amini et al. investigated coretractable modules, and in this work, we obtain some new results on coretractable modules. We prove that every module over a left Artinian left H-ring is coretractable. Furthermore, we show that every module $_RM$ of finite length over a commutative ring $R$ is coretractable. Recall that we say an $R$-module $M$ is co-finite-retractable (simply CFR) whenever for every proper finitely generated submodule $K$ of $M$, Hom$_R(M/K,M)\neq 0$. We show that every CFR module is coretractable if and only if every module is coretractable. Moreover, we show that several of the module classes which we study, are equivalent over a left Noetherian ring. We prove that if $R$ is a left Noetherian ring, then $R$ is left semi-Artinian, and left H-ring if and only if every cyclic $R$-module is coretractable (CFR).

Abstract.Ziegler’s Indecomposability Criterion is used to prove that a totally transcendental, i.e., ∑-pure injective, indecomposable left module over a left noetherian ring is a directed union of finitely generated indecomposable modules. The same criterion is also used to give a sufficient condition for a pure injective indecomposable module RU to have an indecomposable local dual

A result of Ginn and Moss asserts that a left and right noetherian ring with essential right socle is left and right artinian. There are examples of right finitely embedded rings with ACC on left and right annihilators which are not artinian. Motivated by this, it was shown by Faith that a commutative, finitely embedded ring with ACC on annihilators (and square-free socle) is artinian (quasi-Frobenius). A ring R is called right minsymmetric if, whenever k R is a simple right ideal of R, then R k is also simple. In this paper we show that a right noetherian right minsymmetric ring with essential right socle is right artinian. As a consequence we show that a ring is quasi-Frobenius if and only if it is a right and left mininjective, right finitely embedded ring with ACC on right annihilators. This extends the known work in the artinian case, and also extends Faith's result to the non-commutative case.

We study a one-dimensional analogue of representation-finite rings. For a left Noetherian semilocal ring R, we define an R-lattice to be a finitely generated R-module with zero socle. We call R lattice-finite if the number of isomorphism classes of indecomposable R-lattices is finite. Under this assumption, we give several equivalent criteria for the existence of Auslander–Reiten sequences in the category of R-lattices. A necessary condition is that the maximal left quotient ring of R is semisimple, and the main sufficient criterion states that R admits a semiperfect semiprime Asano left overorder.

Criteria are given for a ring to have a left Noetherian largest left quotient ring. It is proved that each such a ring has only finitely many maximal left denominator sets. An explicit description of them is given. In particular, every left Noetherian ring has only finitely many maximal left denominator sets.