The Morphic property in modules and near-rings

Abstract

We introduce and study weakly-morphic modules and their properties. In particular, we show that a finitely generated $\Bbb Z$-module is weakly-morphic if and only if it is finite. Hence a finitely generated Abelian group is morphic if and only if it is weakly-morphic as a $\Bbb Z$-module and each of its primary components is of the form $(\Bbb Z/p^k\Bbb Z)^n$ for some non-negative integers $n$ and $k$. Using these weakly-morphic modules, different notions of a regular module are characterised. We show that, under some special conditions, weakly-morphic property on reduced (respectively, co-reduced) (cyclic) sub-modules reveals the kind of regularity a module will have. Lastly, we study left-morphic near-ring elements and show that the class of left-morphic regular near-rings is properly contained between the classes of left strongly regular and unit-regular near-rings.