Geodesic
Unique Addition Rings
Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 1455-1461

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
Stephenson, W. Unique Addition Rings. Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 1455-1461. doi: 10.4153/CJM-1969-159-0
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