D3-MODULES VERSUS D4-MODULES – APPLICATIONS TO QUIVERS

A module M is called a D4-module if, whenever A and B are submodules of M with M = A ⊕ B and f : A → B is a homomorphism with Imf a direct summand of B, then Kerf is a direct summand of A. The class of D4-modules contains the class of D3-modules, and hence the class of semi-projective modules, and so the class of Rickart modules. In this paper we prove that, over a commutative Dedekind domain R, for an R-module M which is a direct sum of cyclic submodules, M is direct projective (equivalently, it is semi-projective) iff M is D3 iff M is D4. Also we prove that, over a prime PI-ring, for a divisible R-module X, X is direct projective (equivalently, it is Rickart) iff X ⊕ X is D4. We determine some D3-modules and D4-modules over a discrete valuation ring, as well. We give some relevant examples. We also provide several examples on D3-modules and D4-modules via quivers.

On Modules Satisfying the Descending Chain Condition on Cyclic Submodules

A module satisfying the descending chain condition on cyclic submodules is called coperfect. The class of coperfect modules lies properly between the class of locally artinian modules and the class of semiartinian modules. Let R be a commutative ring with identity. We show that every semiartinian R-module is coperfect if and only if R is a T-ring. It is also shown that the class of coperfect R-modules coincides with the class of locally artinian R-modules if and only if 𝔪/𝔪2 is a finitely generated R-module for every maximal ideal 𝔪 of R.

Two Notes on Automorphisms of Modules over a PID

Let D be a principal ideal domain (PID) and M be a module over D. We prove the following two dual results: (i) If M is finitely generated and x, y are two elements in M such that [Formula: see text], then there exists an automorphism α of M such that [Formula: see text]. (ii) If M satisfies the minimal condition on submodules and X, Y are two locally cyclic submodules of M such that [Formula: see text] and [Formula: see text], then there exists an automorphism α of M such that [Formula: see text].

Doily as Subgeometry of a Set of Nonunimodular Free Cyclic Submodules

In this paper, it is shown that there exists a particular associative ring with unity of order 16 such that the relations between non-unimodular free cyclic submodules of its two-dimensional free left module can be expressed in terms of the structure of the generalized quadrangle of order two. Such a doily-centered geometric structure is surmised to be of relevance for quantum information.

Cyclic covering of a module over an Artinian ring

Given a commutative ring with identity [Formula: see text] and an [Formula: see text]-module [Formula: see text], a subset [Formula: see text] of [Formula: see text] is a cyclic covering of [Formula: see text], if this module is the union of the cyclic submodules [Formula: see text], where [Formula: see text]. Such covering is said to be irredundant, if no proper subset of [Formula: see text] is a cyclic covering of [Formula: see text]. In this work, an irredundant cyclic covering of [Formula: see text] is constructed for every Artinian commutative ring [Formula: see text]. As a consequence, a cyclic covering of minimal cardinality of [Formula: see text] is obtained for every finite commutative ring [Formula: see text], extending previous results in the literature.

A NOTE ON SELF-SMALL MODULES OVER RM-DOMAINS

This paper provides two new characterizations of RM-domains, i.e. domains satisfying the restricted minimum condition. A Noetherian domain is a RM-domain if and only if every torsion module is a direct sum of submodules whose cyclic submodules have finite length and homogeneous composition series. We show that this occurs exactly if all self-small torsion modules are finitely generated.