The strong nil-cleanness of semigroup rings

Abstract

In this paper, we study the strong nil-cleanness of certain classes of semigroup rings. For a completely 0-simple semigroup M=ℳ0(G;I,Λ;P)M={ {\mathcal M} }^{0}(G;I,\text{Λ};P), we show that the contracted semigroup ring R0[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;Sα,φα,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[Sα]R{[}{S}_{\alpha }] is strongly nil-clean for each α∈Y\alpha \in Y.

Keywords

• strong nil-cleanness
• semigroup rings
• maximal subgroups
• completely 0-simple semigroups
• supplementary semilattice sum
• 13a02
• 16s34
• 16g30
• 20m25
• 17c17