On \(g(x)\)-invo clean rings. (English) Zbl 1454.16041
Summary: An element in a ring \(R\) with identity is called invo-clean if it is the sum of an idempotent and an involution and \(R\) is called invoclean if every element of \(R\) is invo-clean. Let \(C(R)\) be the center of a ring \(R\) and \(g(x)\) be a fixed polynomial in \(C(R)[x]\). We introduce the new notion of \(g(x)\)-invo clean. \(R\) is called \(g(x)\)-invo if every element in \(R\) is a sum of an involution and a root of \(g(x)\). In this paper, we investigate many properties and examples of \(g(x)\)-invo clean rings. Moreover, we characterize invo-clean as \(g(x)\)-invo clean rings where \(g(x)=(x-a)(x-b),a,b\in C(R)\) and \(b-a\in Inv(R)\). Finally, some classes of \(g(x)\)-invo clean rings are discussed.
MSC:
| 16U60 | Units, groups of units (associative rings and algebras) |
| 16D60 | Simple and semisimple modules, primitive rings and ideals in associative algebras |
| 16S50 | Endomorphism rings; matrix rings |
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