Relationship of Essentially Small Quasi-Dedekind Modules with Scalar and Multiplication Modules

Let be a ring with 1 and D is a left module over . In this paper, we study the relationship between essentially small quasi-Dedekind modules with scalar and multiplication modules. We show that if D is a scalar small quasi-prime -module, thus D is an essentially small quasi-Dedekind -module. We also show that if D is a faithful multiplication -module, then D is an essentially small prime -module iff is an essentially small quasi-Dedekind ring.

ON RICKART MODULES

Gangyong Lee, S.Tariq Rizvi, and Cosmin S.Roman studied Rickart modules. The main purpose of this paper is to develop the properties of Rickart modules . We prove that each injective and prime module is a Rickart module. And we give characterizations of some kind of rings in term of Rickart modules.

SIFAT KEPRIMAAN MODUL SEDERHANA CHEN UNTUK GRAF 𝐴∞

Let 𝐾𝐾 be a field, 𝐸𝐸 is a directed graph. Let 𝐴𝐴~ is a directed line graph. Suppose that 𝑉𝑉[𝑝𝑝] is a class of Chen simple module for the Leavitt path algebra (𝐿𝐿𝐾𝐾 (𝐸𝐸)), with [p] being equivalent classes containing an infinite path. An infinite path p is an infinite sequence from the sides of a graph. In this paper it will be shown that 𝑉𝑉[𝑝𝑝]is not a prime module of the Leavitt path algebra for graph 𝐴𝐴∞ .Keywords : Leavitt path algebra, Graph 𝐴𝐴~, Chen simple modules, Prime modules

The set of singular elements of a ring with respect to a module

Let [Formula: see text] be a ring and [Formula: see text] be a right [Formula: see text]-module. In this paper, we introduce the set [Formula: see text] for some essential submodule [Formula: see text] of [Formula: see text] of singular elements of[Formula: see text] with respect to[Formula: see text] , and we investigate the properties of it. For example, it is shown that [Formula: see text] is an ideal of [Formula: see text] and [Formula: see text]. Also if [Formula: see text] is a semiprime right Goldie ring, then [Formula: see text], where [Formula: see text] is the right singular ideal of [Formula: see text]. We prove that if [Formula: see text] is a semisimple module or a prime module, then [Formula: see text]. For any submodule [Formula: see text] of [Formula: see text], we have [Formula: see text] and if [Formula: see text], then [Formula: see text]. We show that [Formula: see text] and [Formula: see text]. In the end, the singular elements of some rings with respect to the formal triangular matrix ring are investigated.

Prime Modules

Characterizations for prime and semi-prime rings satisfying the right quotient conditions (see § 1) have been determined by A. W. Goldie in (4 and 5). A ring R is prime if and only if the right annihilator of every non-zero right ideal is zero. A natural generalization leads one to consider right R-modules having the properties that the annihilator in R of every non-zero submodule is zero and regular elements in R annihilate no non-zero elements of the module. This is the motivation for the definition of prime module in § 1.