Rings with unique addition

Introduction. The ring { R; +, } is said to have unique addition if there exists no other ring {R; +', } having the same multiplicative semigroup { R; * }. If {R; +, } and {R; +', } are different rings, then the 1-1 mapping 0: aO = a, aER, of { R; +, *} onto { R; +', -} is multiplicative but not additive. Conversely, if there exists a 1-1 mapping 0 of ring {R; +, } onto ring {S; +', } that is multiplicative but not additive, then the ring R does not have unique addition. For we need only define +' on R by: a+'b= (aO+'b6)6-' to obtain a new addition operation on R. Thus, it is clear that every 1-1 multiplicative mapping of ring R onto some ring S is additive if and only if R has unique addition. Rickart [1] has shown that a semi-simple' ring satisfying certain minimum conditions has unique addition. We shall extend Rickart's results to a larger class of rings with minimum conditions in this paper. We have not been able to find any general results for rings without minimum conditions.