It is proven that each indecomposable injective module over a valuation domain R is polyserial if and only if each maximal immediate extension R̂ of R is of finite rank over the completion R̃ of R in the R-topology. In this case, for each indecomposable injective module E, the following invariants are finite and equal: its Malcev rank, its Fleischer rank and its dual Goldie dimension. Similar res...

A module $$M$$ is said to be distributive (resp., uniserial) if the submodule lattice of a chain) Any uniserial but ring integers non-uniserial as $$\mathbb{Z}$$ -module. Direct sums (resp. modules are called semidistributive serial) modules. If $$A$$ with automorphism $$\varphi$$ , then we denote by $$A((x,\varphi))$$ skew Laurent series coefficient in which addition naturally defined and mult...

Let M be an infinite unitary module over a commutative ring R with identity. Then M is called a Jónsson module provided every proper submodule of M has smaller cardinality than M. These modules have been studied by several algebraists, including Robert Gilmer, Bill Heinzer, and the author. In this note, we recall the major results on Jónsson modules to bring the reader up to speed on current re...

A module M over an associative ring R with unity is a QTAG-module if every finitely generated submodule of any homomorphic image of M is a direct sum of uniserial modules.Over the past several years QTAG-modules have been the subject of intense investigation yet there is much to explore.The impetus for these efforts stems from the fact that the rings considered here are almost restriction free....

In the study of hereditary Noetherian rings, it is clear that hereditary Noetherian prime rings will play a central role (see, for example, [12]). Here we study the (two-sided) ideals of an hereditary Xoetherian prime ring and, as a consequence, ascertain the structure of factor rings and torsion modules. The torsion theory represents a generalization of similar results about Dedekind prime rin...

It is proved that if R is a valuation domain with maximal ideal P and if RL is countably generated for each prime ideal L, then R R is separable if and only RJ is maximal, where J = ∩n∈NP . When R is a valuation domain satisfying one of the following two conditions: (1) R is almost maximal and its quotient field Q is countably generated (2) R is archimedean Franzen proved in [2] that R is separ...

Let R be a commutative ring. An R-module M is called co-multiplication provided that foreach submodule N of M there exists an ideal I of R such that N = (0 : I). In this paper weshow that co-multiplication modules are a generalization of strongly duo modules. Uniserialmodules of finite length and hence valuation Artinian rings are some distinguished classes ofco-multiplication rings. In additio...

A new construction is given of non-standard uniserial modules over certain valuation domains; the construction resembles that of a special Aronszajn tree in set theory. A consequence is the proof of a sufficient condition for the existence of non-standard uniserial modules; this is a theorem of ZFC which complements an earlier independence result.

We apply minimal weakly generating sets to study the existence of Add ( U R ) -covers for a uniserial module . If is right over ring , then S : = End has at most two maximal (right, left, two-sided) ideals: one set I all endomorphisms that are not injective, and other K surjective. prove if either finitely generated, or artinian, ? class covering only it closed under direct limit. Moreover, we ...

It is proved that localizations of injective R-modules of finite Goldie dimension are injective if R is an arithmetical ring satisfying the following condition: for every maximal ideal P , RP is either coherent or not semicoherent. If, in addition, each finitely generated R-module has finite Goldie dimension, then localizations of finitely injective R-modules are finitely injective too. Moreove...