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The Singular Congruence and the Maximal Quotient Semigroup

Published online by Cambridge University Press:  20 November 2018

F. R. McMorris
F. R. McMorris*
Affiliation:
Bowling Green State University, Bowling Green, Ohio
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It is a well known result (see [4, p. 108]) that if R is a ring and Q(R) its maximal right quotient ring, then Q(R) is (von Neumann) regular if and only if every large right ideal of R is dense. This condition is equivalent to saying that the singular ideal of R is zero. In this note we show that the condition loses its magic in the theory of semigroups.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 15 , Issue 2 , June 1972 , pp. 301 - 303
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Clifford, A. H. and Preston, G. B., The algebraic theory of semigroups, Amer. Math. Soc. Surveys, No. 7, Vol. I, Providence, R.I., 1961.Google Scholar
2. Clifford, A. H. and Preston, G. B., The algebraic theory of semigroups, Amer. Math. Soc. Surveys, No. 7, Vol. H, Providence, R.I., 1967.Google Scholar
3. Johnson, R. E., The extended centralizer of a ring over a module, Proc. Amer. Math. Soc. 2 (1951), 891-895.Google Scholar
4. Lambek, J., Lectures on rings and modules, Blaisdell, Waltham, Mass., 1966.Google Scholar
5. McMorris, F. R., The maximal quotient semigroup, Semigroup Forum (to appear).Google Scholar
6. McMorris, F. R., The quotient semigroup of a semigroup that is a semilattice of groups, Glasgow Math. J. 12, (1971), 18-23.Google Scholar