On rings with commuting ideals

This report is a summarization and extension of previous work done by Dr. Efraim Armendariz, University of Texas at Austin, and Dr. Henry E. Heatherly, University of Lousiana Lafayette, on the topic of rings with commuting ideals. Some of these authors’ results are extended to one-sided ideals instead of two-sided ideals. Topics discussed include homomorphic images of rings with commuting left ideals, finite direct products of rings with commuting left ideals, rings of n x n matrices over a ring with commuting left ideals, constructing rings with commuting ideals using an idempotent, and simple rings with commuting ideals over polynomials in X. Examples are given to illustrate some properties of rings with commuting ideals. A short a discussion regarding the inclusion of ring theory in the secondary mathematics classroom is also included.