Left-Right Cleanness and Nil Cleanness
in Unital Rings
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.
Peter Vassilev Danchev, Ph.D. (Math.), Prof., Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev st., bl. 8, Sofia, 1113, Bulgaria, e-mail: email@example.com; firstname.lastname@example.org
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