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Regular, Commutative, Maximal Semigroups of Quotients

Published online by Cambridge University Press:  20 November 2018

Jurgen Rompke
Jurgen Rompke*
Affiliation:
McMaster University, Hamilton, Ontario, Canada8000 München 40 Germania STR. 5 Fed. Rep. of Germany
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A well-known theorem which goes back to R. E. Johnson [4], asserts that if R is a ring then Q(R), its maximal ring of quotients is regular (in the sense of v. Neumann) if and only if the singular ideal of R vanishes. In the theory of semigroups a natural question is therefore the following: Do there exist properties which characterize those semigroups whose maximal semigroups of quotients are regular? Partial answers to this question have been given in [3], [7] and [8]. In this paper we completely solve the commutative case, i.e. we give necessary and sufficient conditions for a commutative semigroup S in order that Q(S), the maximal semigroup of quotients, is regular. These conditions reflect very closely the property of being semiprime, which in the theory of commutative rings characterizes those rings which have a regular ring of quotients.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 18 , Issue 1 , April 1975 , pp. 99 - 104
Copyright
Copyright © Canadian Mathematical Society 1975

References

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