Trivial Extension of π-Regular Rings | ||
| Engineering and Technology Journal | ||
| Article 1, Volume 34, 1B, January 2016, Pages 153-159 PDF (617.24 K) | ||
| DOI: 10.30684/etj.34.1B.17 | ||
| Author | ||
| Areej M. Abduldaim | ||
| Abstract | ||
| In this paper we investigate if it is possible that the trivial extension ring T(R,R) inherit the properties of the ring R and present the relationship between the trivial extension T(R,M) of a ring R by an R-module M and theπ-regularity of Rby taking new concepts asπ-coherent rings and C-π-regular rings whichintroducedas extensions of the concept of π-regularrings. Moreover we studied the possibility of being the trivial extensionT(R,M) itselfπ-regular ring according to specific conditions. Thus we proved that if R is an Artinian ring, then the trivial extension T(R,R) is a π-regular ring. As well as ifthe trivial extension T(R,R) is a Noetherian π-regularring, then R is a π-regular ring. On the other hand we showed that if F is a field and M is an F-vector space with infinite dimension, then the trivial extension ring T(F,M) of F by M is C-π-regular ring. | ||
| Keywords | ||
| Trivial extension ring; regular ring; Artinian ring; Noetherian ring | ||
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