Lifting Idempotents in Associative Pairs

AbstractIn 1977, Nicholson developed the theory of suitable rings (Trans. Amer. Math. Soc.229 (1977), 269–278), namely, those in which idempotents lift modulo every left (right) ideal. In this paper the concepts of lifting idempotents modulo every left ideal in an associative pair (A+,A−) and lifting (von Neumann) regular elements modulo every left ideal of A+ (resp. A−) are introduced and shown to be equivalent. We study the behavior of a pair and its standard embedding with respect to the property of being idempotent-lifting. It is also proved that the Jacobson radical can be characterized as the largest one-sided ideal containing no nonzero regular elements. Finally, we show that lifting orthogonal idempotents is possible in this class of pairs