Quasi-clean rings and strongly quasi-clean rings
| Autor: | Gaohua Tang, Pingzhi Yuan, Huadong Su |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
Pure mathematics
Mathematics::Commutative Algebra Computer Science::Information Retrieval Applied Mathematics General Mathematics Central unit Idempotence Astrophysics::Instrumentation and Methods for Astrophysics Computer Science::General Literature Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing) Element (category theory) Ring (chemistry) Mathematics |
| Zdroj: | Communications in Contemporary Mathematics. 25 |
| ISSN: | 1793-6683 0219-1997 |
| DOI: | 10.1142/s0219199721500796 |
| Popis: | An element [Formula: see text] of a ring [Formula: see text] is called a quasi-idempotent if [Formula: see text] for some central unit [Formula: see text] of [Formula: see text], or equivalently, [Formula: see text], where [Formula: see text] is a central unit and [Formula: see text] is an idempotent of [Formula: see text]. A ring [Formula: see text] is called a quasi-Boolean ring if every element of [Formula: see text] is quasi-idempotent. A ring [Formula: see text] 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 [Formula: see text] has no image isomorphic to [Formula: see text]; For an indecomposable commutative semilocal ring [Formula: see text] with at least two maximal ideals, [Formula: see text]([Formula: see text]) is strongly quasi-clean if and only if [Formula: see text] is quasi-clean if and only if [Formula: see text], [Formula: see text] is a maximal ideal of [Formula: see text]. For a prime [Formula: see text] and a positive integer [Formula: see text], [Formula: see text] is strongly quasi-clean if and only if [Formula: see text]. Some open questions are also posed. |
| Databáze: | OpenAIRE |
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