Left-right cleanness and nil cleanness in unital rings

P. V. Danchev

Institute of Mathematics and Informatics of Bulgarian Academy of Sciences, Sofia, Bulgaria

Abstract: We introduce the notions of left and right cleanness and nil cleanness in rings showing their close relationships with the classical concepts of cleanness and nil cleanness. Specifically, it is proved that strongly clean rings are both L-clean and R-clean as well as strongly nil clean rings are both L-nil clean and R-nil clean. These two assertions somewhat strengthen well-known results due to Nicholson (Comm. Algebra, 1999) and Diesl (J. Algebra, 2013). Moreover, it is shown that L-nil cleanness (respectively, R-nil cleanness) is preserved modulo nil Jacobson radical as well as that this is still true for L-cleanness (respectively, R-cleanness), provided the Jacobson radical is nil.

Keywords: clean rings, nil clean rings, L-clean rings, R-clean rings, L-nil clean rings, R-nil clean rings.

UDC: 512.552.13

MSC: 16U99, 16E50, 13B99

Received: 08.12.2018

Language: English

DOI: 10.26516/1997-7670.2019.27.28

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