The strong nil-cleanness of semigroup rings

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 .