Feckly Adequate Conditions and Elementary Matrix Reduction

TL;DR

This paper introduces new conditions under which Bezout rings are elementary divisor rings, establishing equivalences involving feckly zero-adequacy and regularity, and generalizing several known results to broader classes of rings.

Contribution

It establishes that feckly zero-adequate rings are elementary divisor rings and characterizes elementary divisor rings with feckly adequate range 1, broadening existing theorems.

Findings

Feckly zero-adequate rings are elementary divisor rings.

Equivalence between feckly zero-adequacy and regularity of R/J(R).

Characterization of elementary divisor rings with feckly adequate range 1.

Abstract

We present some new conditions for a B$\acute{e}$zout ring to be an elementary divisor ring. We prove, in this note, that a B$\acute{e}$zout ring $R$ is feckly zero-adequate if and only if $R/J(R)$ is regular if and only if $R/J(R)$ is $\pi$-regular, and that every feckly zero-adequate ring is an elementary divisor ring. If $R$ has feckly adequate range 1, we prove that $R$ is an elementary divisor ring if and only if $R$ is a B$\acute{e}$zout ring. Many known results are thereby generalized to much wider class of rings, e.g. [4, Theorem 14], [5, Theorem 4], [8, Theorem 1.2.14], [10, Theorem 4] and [11, Theorem 7]. \vskip3mm {\bf Keywords:} Elementary divisor ring, B$\acute{e}$zout ring, Feckly zero-adequate ring, Feckly adequate range 1.