PIECEWISE SEMIPRIME RINGS AND SOME APPLICATIONS

Year 2014, Volume: 15 Issue: 15, 26 - 40, 01.06.2014

Mehdi Gurabi Ahmad Haghany Mohammad Reza Vedadi

https://doi.org/10.24330/ieja.266235

https://izlik.org/JA22HG99LZ

Abstract

We define piecewise semiprime (PWSP) rings R in terms of a set
of triangulating idempotents in R. The class of PWSP rings properly contains
both the class of semiprime rings and the class of piecewise prime rings. The
PWSP property is Morita invariant and it is shared by some important ring
extensions. A ring is PWSP if and only if it has a generalized upper triangular
matrix representation with semiprime rings on the main diagonal. Another
characterization of PWSP rings involves a generalization of the concept of
m-systems and is similar to the description of a semiprime ring in terms of
the prime radical. Finally we use the PWSP property to determine (right)
weak quasi-Baer rings. These are rings in which the right annihilator of every
nilpotent ideal is generated as a right ideal by an idempotent.

Keywords

Piecewise prime , piecewise semiprime , triangulating idempotents , weak quasi-Baer ring