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Semigroups and rings whose proper one-sided ideals are power joined. (English) Zbl 0543.20042
A semigroup S is said to be power joined if for any a,\(b\in S\) there exist positive integers m,n such that \(a^ m=b^ n\). In this paper the authors prove the following: Theorem 1.6. Every proper one-sided ideal of a semigroup S is power joined if and only if S satisfies one of the following conditions: (i) S is power joined, (ii) S is a periodic group, (iii) S is a left (right) zero-semigroup of two periodic groups, and (iv) S is a semilattice of two semigroups M and \(S\backslash M\), where M is power joined and coincides with the greatest ideal of S, and \(S\backslash M\) is a group. Moreover, the identity of \(S\backslash M\) is the identity for S. Theorem 2.1. Every proper one-sided ideal of a ring R is multiplicatively power joined if and only if R satisfies one of the following conditions: (i) R is a nilring, and (ii) R is a ring with identity and (R,.) is a semilattice of two semigroups M and \(R\backslash M\), where M is a nilring and coincides with the greatest ideal of R, and \(R\backslash M\) is a group.
Reviewer: B.Pondelíček
MSC:
20M10 General structure theory for semigroups
20M12 Ideal theory for semigroups
16N40 Nil and nilpotent radicals, sets, ideals, associative rings
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References:
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