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The semigroup of endomorphisms of a Boolean ring

Published online by Cambridge University Press:  09 April 2009

Kenneth D. Magill Jr
Kenneth D. Magill Jr
Affiliation:
State University of New York at Buffalo
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The family (R) of all endomorphisms of a ring R is a semigroup under composition. It follows easily that if R and T are isomorphic rings, then (R) and (T) are isomorphic semigroups. We devote ourselves here to the converse question: ‘If (R) and (T) are isomorphic, must R and T be isomorphic?’ As one might expect, the answer is, in general, negative. For example, the ring of integers has precisely two endomorphisms – the zero endomorphism and the identity automorphism. Since the same is true of the ring of rational numbers, the two endomorphism semigroups are isomorphic while the rings themselves are certainly not.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 11 , Issue 4 , November 1970 , pp. 411 - 416
Copyright
Copyright © Australian Mathematical Society 1970

References

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