let $ m , n in mathbb{n}$, $d$ be a division ring, and $m_{m times n}(d)$ denote the bimodule of all $m times n$ matrices with entries from $d$. first, we characterize one-sided submodules of $m_{m times n}(d)$ in terms of left row reduced echelon or right column reduced echelon matrices with entries from $d$. next, we introduce the notion of a nest module of matrices with entries from $d$. we ...

A method for solving CCS equations of type (AIX)\L ~ B, where X is unknown, is presented. The method is useful in a top-down design methodology: if a system (B) and some of its submodules (.4) axe specified, solving such an equation amounts to constructing the missing submodules. The method works by successively transforming equations into simpler equations, in parallel with generation of a sol...

Let [Formula: see text] be a commutative ring and nonzero text]-module. We introduce the class of pseudo-strongly (PS)-hollow submodules text]. Inspired by theory modules with secondary representations, we investigate which can written as finite sums PS-hollow submodules. In particular, provide existence uniqueness theorems for minimal strongly representations over Artinian rings.

In this note we show that a Noetherian module has a dual module if and only if it satisfies AB5*. A connection between completeness and AB5* is also established. In this note we relate completeness, quasi-completeness, the A B5* condition, and duality. The main result is that a Noetherian R-module has a dual module if and only if it satisfies A B5*. Throughout this note R will denote a commutat...

Let R be a commutative ring with an identity. The purpose of this paper is to introduce and investigate the notion strong id="M3"> z ° -submodules id="M4"> -module as extension id="M5"> quasi id="M6"> -submodules.

All rings are commutative with identity and all modules are unital. The tensor product of projective (resp. flat, multiplication) modules is a projective (resp. flat, multiplication) module but not conversely. In this paper we give some conditions under which the converse is true. We also give necessary and sufficient conditions for the tensor product of faithful multiplication Dedekind (resp. ...