COMPUTATIONAL ALGORITHM FOR AN MP3 (MOORE-PENROSE 3) MATRIX RING AND FORMULATION OF MP3 EQUIVALENCE

Abstract

The structure theorems for complete local rings are largely due to Irving S. Cohen. The beginning part of Cohen's structure theorem for the equicharacteristic case is A complete local ring which has the same characteristic as its residue field contains a coefficient field, see (1) The second part of Cohen's structure theorem conveys essentially that in the equicharacteristic case, a complete local ring is a homormorphic image of a formal power series ring over the residue field. This paper proposes the algebraic structure of an n × n matrix ring Mn(R) (n ≥ 1) where R is a formal power series in one variable over a field modulo a power of its maximal ideal. The structure of this special class of matrix rings is derived using the notion of a pseudoinverse and a generalization of one of the Moore-Penrose conditions. Afterwards, some theoretical properties are given to establish the equivalence of MP3 (Moore-Penrose 3) rings that are becoming more exciting in theoretical algebraic investigations for their symmetry inducing capability.