Abstract
Abstract In this paper, we study the strong nil-cleanness of certain classes of semigroup rings. For a completely 0-simple semigroup M = ℳ 0 ( G ; I , Λ ; P ) M={ {\mathcal M} }^{0}(G;I,\text{Λ};P) , we show that the contracted semigroup ring R 0 [ M ] {R}_{0}{[}M] is strongly nil-clean if and only if either | I | = 1 |I|=1 or | Λ | = 1 |\text{Λ}|=1 , and R [ G ] R{[}G] is strongly nil-clean; as a corollary, we characterize the strong nil-cleanness of locally inverse semigroup rings. Moreover, let S = [ Y ; S α , φ α , β ] S={[}Y;{S}_{\alpha },{\varphi }_{\alpha ,\beta }] be a strong semilattice of semigroups, then we prove that R [ S ] R{[}S] is strongly nil-clean if and only if R [ S α ] R{[}{S}_{\alpha }] is strongly nil-clean for each α ∈ Y \alpha \in Y .
Highlights
Diesl [1] introduced the concept of nil-clean and strongly nil-clean rings and asked the question when a matrix ring is nil-clean
By [1], if R is a commutative ring with identity, the formal block matrix ring is strongly nil-clean if and only if are strongly nil-clean
General contracted completely 0-simple semigroup rings can be viewed as a generalization of matrix rings
Summary
Diesl [1] introduced the concept of nil-clean and strongly nil-clean rings and asked the question when a matrix ring is (strongly) nil-clean. General contracted completely 0-simple semigroup rings can be viewed as a generalization of matrix rings. For a finite completely 0-simple semigroup M = 0(G;I, Λ;P), the contracted semigroup ring R0[M] contains an identity if and only if |I| = |Λ| and P is an invertible matrix over G0, and if and only if R0[M] is isomorphic to the matrix ring |I|(R[G]) [6]. Throughout this paper, a ring always means an associate non-unital (or general) ring, and we always assume that R is a ring with identity (not necessarily commutative)
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