On matrix rings with the SIP and the Ads

TAKIL MUTLU F.

TURKISH JOURNAL OF MATHEMATICS, vol.42, no.5, pp.2657-2663, 2018 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 42 Issue: 5
  • Publication Date: 2018
  • Doi Number: 10.3906/mat-1609-77
  • Journal Name: TURKISH JOURNAL OF MATHEMATICS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, TR DİZİN (ULAKBİM)
  • Page Numbers: pp.2657-2663
  • Keywords: Ads property, summand intersection property, trivial extension, SUMMAND INTERSECTION PROPERTY, MODULES
  • Anadolu University Affiliated: Yes

Abstract

In this paper, matrix rings with the summand intersection property (SIP) and the absolute direct summand (ads) property (briefly, SA) are studied. A ring R has the right SIP if the intersection of two direct summands of R is also a direct summand. A right R-module M has the ads property if for every decomposition M = A circle plus B of M and every complement C of A in M, we have M = A circle plus C. It is shown that the trivial extension of R by M has the SA if and only if R has the SA, M has the ads, and (1 - e)Me = 0 for each idempotent e in R. It is also shown with an example that the SA is not a Morita invariant property.