# IJPAM: Volume 45, No. 2 (2008)

**STRUCTURE THEOREMS AND STATISTICAL APPLICATION**

FOR MATRIX RINGS OVER MOORE-PENROSE

TWO (MP2) RINGS

FOR MATRIX RINGS OVER MOORE-PENROSE

TWO (MP2) RINGS

Department of Mathematics

Grambling State University

Grambling, Louisiana, 71245, USA

**Abstract.**The mathematicians Edwin Moore [#!1!#] and Roger Penrose [#!2!#] authored the
Moore-Penrose conditions which assert that given any nonzero matrix A over
the complex field, there exists a nonzero matrix X such that: (1) AXA = A; (2) XAX = X; (3) (XA) = XA; (4) (AX)= AX. This paper generalizes the second Moore-Penrose condition to an
arbitrary ring R which will be called MP2 as follows: Given any nonzero
element a in R, there exists a nonzero x in R such that xax = x.
Accordingly, the structure theorems for such MP2 rings are developed, as
well as the structure theorems for matrix rings over them. Interestingly
enough, MP2 rings appear frequently in physical chemistry for converting
linear operators to symmetric ones, and in engineering applications for
solving unstable linear systems, or in business demand-supply matrix models
with ill-conditioned Leontif matrices.

**Received: **March 28, 2008

**AMS Subject Classification: **17C27

**Key Words and Phrases: **idempotent, principal left ideal, matrix ring, annihilator

**Source:** International Journal of Pure and Applied Mathematics

**ISSN:** 1311-8080

**Year:** 2008

**Volume:** 45

**Issue:** 2