# zbMATH — the first resource for mathematics

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
##### Keywords:
power joined; zero-semigroup; periodic groups; semilattice; nilring
Full Text:
##### References:
  Cherubini A., A. Varisco: On Putcha’s Q-semigroups. Semigroup Forum 18 (1979) 313-317. · Zbl 0429.20051  Clifford A. H., G. B. Preston: The algebraic theory of semigroups. Amer. Math. Soc., Vol. I (1961). · Zbl 0111.03403  Pondělíček B.: Uniform semigroups whose proper quasi-ideals are power joined. Semigroup Forum 22 (1981) 331-337. · Zbl 0465.20058  Petrich M.: Introduction to semigroups. Merrill, Columbus (1973). · Zbl 0321.20037  Putcha M. S.: Band of r-archimedean semigroups. Semigroup Forum 6 (1973) 232-239. · Zbl 0262.20070  Putcha M. S.: Rings which are semilattices of archimedean semigroups. Semigroup Forum 23 (1981) 1-5. · Zbl 0469.20038  Cherubini A. A. Varisco: Sui semigruppi i cui sottosemigruppi proprî sono t-archimedei. Istituto Lombardo (Rend. Sc.) 112 (1978) 91-98. · Zbl 0434.20037  Nagy A.: Semigroups whose proper two-sided ideals are power joined. Semigroup Forum 25 (1982) 325-329. · Zbl 0498.20047
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.