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Unique Addition Rings

Published online by Cambridge University Press:  20 November 2018

W. Stephenson
W. Stephenson*
Affiliation:
Bedford College, Regent's Park, London
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A semigroup (R, ⋅) is said to be a unique addition ring (UA-ring) if there exists a unique binary operation + making (R, ⋅, + ) into a ring. All our results can be presented in this semigroup theoretic setting. However, we prefer the following equivalent ring theoretic formulation: a ring R is a UA-ring if and only if any semigroup isomorphism α: (R, ⋅) ≅ (S, ⋅) with another ring S is always a ring isomorphism.

UA-rings have been studied in (8; 4) and are also touched on in (1; 2; 6; 7). In this note we generalize Rickart's methods to much wider classes of rings. In particular, we show that, for a ring R with a 1 and n ≧ 2, the (n × n) matrix ring over R and its subring of lower triangular matrices are UA-rings.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 1455 - 1461
Copyright
Copyright © Canadian Mathematical Society 1969

References

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