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Quasi-clean rings and strongly quasi-clean rings.
- Source :
-
Communications in Contemporary Mathematics . Mar2023, Vol. 25 Issue 2, p1-19. 19p. - Publication Year :
- 2023
-
Abstract
- An element a of a ring R is called a quasi-idempotent if a 2 = k a for some central unit k of R , or equivalently, a = k e , where k is a central unit and e is an idempotent of R. A ring R is called a quasi-Boolean ring if every element of R is quasi-idempotent. A ring R is called (strongly) quasi-clean if each of its elements is a sum of a quasi-idempotent and a unit (that commute). These rings are shown to be a natural generalization of the clean rings and strongly clean rings. An extensive study of (strongly) quasi-clean rings is conducted. The abundant examples of (strongly) quasi-clean rings state that the class of (strongly) quasi-clean rings is very larger than the class of (strongly) clean rings. We prove that an indecomposable commutative semilocal ring is quasi-clean if and only if it is local or R has no image isomorphic to ℤ 2 ; For an indecomposable commutative semilocal ring R with at least two maximal ideals, n (R) (n ≥ 2) is strongly quasi-clean if and only if n (R) is quasi-clean if and only if min { | R / m | , m is a maximal ideal of R } > n + 1. For a prime p and a positive integer n ≥ 2 , n (ℤ (p)) is strongly quasi-clean if and only if p > n. Some open questions are also posed. [ABSTRACT FROM AUTHOR]
- Subjects :
- *COMMUTATIVE rings
*ASSOCIATIVE rings
*OPEN-ended questions
Subjects
Details
- Language :
- English
- ISSN :
- 02191997
- Volume :
- 25
- Issue :
- 2
- Database :
- Academic Search Index
- Journal :
- Communications in Contemporary Mathematics
- Publication Type :
- Academic Journal
- Accession number :
- 161829437
- Full Text :
- https://doi.org/10.1142/S0219199721500796