Uniserial Rings Research Articles

A note on self-dual modules and Dedekind rings

It is shown that a commutative noetherian ring is a finite direct sum of Dedekind rings and artinian uniserial rings if and only if every module of finite length is selfdual. A module of finite length is said to be selfdual if it is isomorphic to its dual with respect to the minimal injective cogenerator.

Modules with Indecomposable Decompositions That Complement Maximal Direct Summands

Generalizing a fundamental property of semisimple modules, Anderson w x and Fuller 1 introduced the following important concept for direct decompositions of modules. A decomposition M s [ M is said to ig I i complement direct summands in case for each direct summand A of M, Ž . Ž w there is a subset H : I such that M s A [ [ M cf. also 2, p. ig H i x. 141 . Such a decomposition is necessarily an indecomposable decomposition, and it is an interesting problem to study under which conditions an indecomposable decomposition complements direct summands. There is w x an extensive literature concerning this problem. By 1, Theorem 6 , right perfect rings can be characterized by the property that all projective right modules have decompositions that complement direct summands. Fuller w x 7 showed that over a generalized uniserial ring every module has a decomposition that complements direct summands, thus providing a first class of non-semisimple rings satisfying this property. More generally, w x Ž . Tachikawa 14 proved that every left and right module over a ring of finite representation type has a decomposition that complements direct summands, and that the converse is also true was established by Fuller and w x w x Reiten 9 . Restricting just to one side, Fuller 8 showed that rings over which every right module has a decomposition that complements direct summands are precisely the rings over which every right module is a direct Ž sum of finitely generated modules they are also called right pure-semisim-

A note on generalished uniserial ring

A note on generalished uniserial ring

Balanced modules and rings of left colocal type

is a ring homomorphism, where rm is the left multiplication of r, i.e., rm(x)= rx for all x^M. If p is surjective, or equivalently every element of BiEndG?M) is a left multiplication of an element of R, then M is said to be balanced. If every finitely generated faithfulleft i?-module is balanced, then R is called a left QF-l ring. Further, R is said to be a left balanced ring if R/l is left QF-l for every two sided ideal / of R; this condition is equivalent to the condition that every finitelygenerated left module is balanced. The concept of QF-l rings was introduced by Thrall [18] as a generalization of quasi-Frobenius rings, and in the same paper he proposed to give an internal characterization of QF-l rings. At the present time, however, this problem is not solved completely, though partial answers are given in several cases (Camillo and Fuller [1], Dlab and Ringel [2], Fuller [7], Makino [8, 9, 10], Morita [11], Ringel [12] and Tachikawa [17]). On the other hand, the structure of balanced rings was completely determined by Dlab and Ringel [2, 3, 4, 5, 6]. Indeed they proved that an indecomposable ring is left balanced if and only if it is a full matrix ring over either a local uniserial ring or an exceptional ring. We should note that exceptional rings are characterized as local rings of either left or right colocal type with the square zero radical (see Section 1 for the precise definition of rings of left colocal type). Our aim of this paper is to show that in the case of left serial rings, left QF-lness, which is much more weaker condition than left balancedness, implies rings of left colocal tvoe (Theorem). For the case of a finite dimensional

A characterization of uniserial rings via continuous and discrete modules

AbstractIt is well-known that quasi-Frobenius rings are characterized by the property that all propective right modules are injective, as well as by the property that all injective right modules are projective. Similarly, either the property that every quasi-injective or that every quasi-injective is quasi-projective characterizes uniserial rings. Oshiro has given similar characterizations for generalized uniserial rings. The purpose of this paper is to characterize rings for which continuous right modules are discrete. We show that these rings are precisely the uniserial rings. The property that every discrete module is continuous is also investigated.

Rings whose cyclics are essentially embeddable in projective modules

A ring R is quasi-Frobenius (QF) if and only if every right R-module is embeddable in a projective module. We call a ring R a right (left) CEPring if each cyclic right (left) R-module is essentially embeddable in a projective module. Examples of right CEP-rings include QF-rings and right uniserial rings. Indeed R is a QF-ring if and only if R is both a right and left CEP-ring [S]. A right CEP-ring which is QF-3 is shown to be QF (Theorem 3.3). Semiperfect CEP-rings and rings, each of whose homomorphic images is a right CEP-ring, are characterized in Theorems 5.2 and 6.2, respectively. The last section deals with split extensions of right uniserial rings as examples of CEP-rings.

A generalization of the Wedderburn-Artin theorem

The structure of rings such that each of its homomorphic images has the property that each cyclic right module over it is essentially embeddable in a direct summand is determined. Such rings are precisely (i) right uniserial rings, (ii) n × n n \times n matrix rings over two-sided uniserial rings with n > 1 n > 1 , or (iii) sums of rings of the types (i) and (ii).

Finite uniserial rings of prime characteristic

Finite uniserial rings of prime characteristic

Rings over which all cyclic modules are poorly injective

The fundamental result of the paper is Theorem I, asserting that over a right Noetherian ring R, the poor injectivity of all cyclic modules is equivalent to the fact that R is a direct sum of a semisimple Artinian ring and a finite number of uniserial rings and hereditary Noetherian left-invariant domains. (We note that the injectivity of all cyclic R-modules over an arbitrary ring R is equivalent to the classical semisimplicity of the ring R [2].)

Uniserial Rings and Skew Polynomial Rings

Uniserial Rings and Skew Polynomial Rings

Serial Right Noetherian Rings

A module M is called a serial module if the family of its submodules is linearly ordered under inclusion. A ring R is said to be serial if RR as well as RR are finite direct sums of serial modules. Nakayama [8] started the study of artinian serial rings, and he called them generalized uniserial rings. Murase [5, 6, 7] proved a number of structure theorems on generalized uniserial rings, and he described most of them in terms of quasi-matrix rings over division rings. Warfield [12] studied serial both sided noetherian rings, and showed that any such indecomposable ring is either artinian or prime. He further showed that a both sided noetherian prime serial ring is an (R:J)-block upper triangular matrix ring, where R is a discrete valuation ring with Jacobson radical J. In this paper we determine the structure of serial right noetherian rings (Theorem 2.11).

Lifting modules, extending modules and their applications to generalized uniserial rings

Lifting modules, extending modules and their applications to generalized uniserial rings

Semi-Prime Rings Whose Homomorphic Images are Serial

A theorem of Eisenbud, Griffith, and Robson states that if R is hereditary and noetherian (on both the left and right) then every proper homomorphic image of R is a generalized unserial ring (see, for example, [3, p. 244]). Singh [11, p. 883] states a converse: If R is a right bounded, noetherian prime ring, all of whose proper homomorphic images are generalized uniserial rings, then (every divisible right R-module is injective, so) R is right hereditary. (Actually, Singh omitted the clearly necessary “bounded” condition.) Singh's theorem generalizes results of [9, Proposition 15], [2, Theorem 2.1], and [8], about commutative rings.We will call a semi-prime ring R essentially right bounded if each essential right ideal contains a two-sided ideal which is essential as a right ideal. In case R is prime, “essentially right bounded” coincides with “right bounded”.

Uniserial rings and lifting properties

Uniserial rings and lifting properties

Stable Equivalence of Uniserial Rings

It is well known that two basic rings are Morita equivalent if and only if they are isomorphic. Here it is shown that local uniserial rings are stably equivalent in the sense of M. Auslander and I. Reiten in case they are isomorphic modulo certain powers of their radicals. In particular, two commutative local uniserial rings of Loewy length n are stably equivalent if and only if they are isomorphic modulo the [n/2]th power of their radicals.

Stable equivalence of uniserial rings

It is well known that two basic rings are Morita equivalent if and only if they are isomorphic. Here it is shown that local uniserial rings are stably equivalent in the sense of M. Auslander and I. Reiten in case they are isomorphic modulo certain powers of their radicals. In particular, two commutative local uniserial rings of Loewy length n are stably equivalent if and only if they are isomorphic modulo the [n/2]th power of their radicals.

Generalized uniserial rings

The following conditions are shown to be equivalent: 1) ring A is generalized uniserial (not necessarily artinian); 2) every finitely presented A module is semiserial; 3) A is semiperfect and the projective cover of every indecomposable finitely presented module is indecomposable.

Rings whose Modules are Projective over Endomorphism Rings

Let $R$ be either an Artin ring or a commutative ring. We show that every $R$-module is projective over its endomorphism ring if and only if $R$ is uniserial. We study rings, all of whose modules are projective over their endomorphism rings. We show that for Artin or commutative rings this is equivalent to being uniserial. We remark that we know of no examples other than uniserial rings and conjecture that all such rings are uniserial. Our proof uses the complete ring of quotients as in [4]. Our work is partially motivated by the work of Sally and Vasconcelos on commutative rings whose ideals are projective over their endomorphism rings [7]. We use $J$ throughout to denote the Jacobson radical and $\operatorname {Soc} (R)$ to denote the socle of $R$.

Left generalized uniserial rings

Left generalized uniserial rings

Nonsingular rings with essential socles

This paper is a study of nonsingular rings with essential socles. These rings were first investigated by Goldie [5] who studied the Artinian case and showed that an indecomposable nonsingular generalized uniserial ring is isomorphic to a full blocked triangular matrix ring over a sfield. The structure of nonsingular rings in which every ideal generated by a primitive idempotent is uniform was determined for the Artinian case by Gordon [6] and Colby and Rutter [2], and for the semiprimary case by Zaks [12]. Nonsingular rings with essential socles and finite identities were characterized by Gordon [7] and the author [10]. All these results were obtained by representing the rings in question as matrix rings. In this paper a matrix representation of arbitrary nonsingular rings with essential socles is found (section 2). The above results are special cases of this representation. A general method for representing rings as matrices is developed in section 1.