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Local semigroup rings. (Russian. English summary) Zbl 0870.16019

Author’s abstract: The description of local semigroup algebras over a field of characteristic \(p\) (if \(p>0\), then semigroups are assumed to be locally finite) due to J. Okniński [Glasg. Math. J. 25, 37-44 (1984; Zbl 0529.20052)] is transferred to semigroup rings over non-radical rings. The following statement is proved. Let \(R\) be a ring, \(R\neq J(R)\), \(\text{char }R=0\) (\(\text{char }R=p>1\)), \(S\) be a semigroup (respectively, a locally finite semigroup). The semigroup ring \(R[S]\) is local [scalar local] if and only if \(R\) is such a ring and \(S\) is an ideal extension of a rectangular band (respectively of a completely simple semigroup over a \(p\)-group) by a locally nilpotent semigroup.

MSC:

16S36 Ordinary and skew polynomial rings and semigroup rings
16L30 Noncommutative local and semilocal rings, perfect rings
20M25 Semigroup rings, multiplicative semigroups of rings

Citations:

Zbl 0529.20052
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