ON RINGS DETERMINED BY ZERO PRODUCTS

Abstract

Let [Formula: see text] be a ring. We say that [Formula: see text] is zero product determined if for every additive group [Formula: see text] and every bi-additive map [Formula: see text] the following holds: if ϕ(a, b) = 0 whenever ab = 0, then there exists an additive map [Formula: see text] such that ϕ(a, b) = T(ab) for all [Formula: see text]. In this paper, first we study the properties of zero product determined rings and show that semi-commutative and non-commutative rings are not zero product determined. Then, we will examine whether the rings with a nontrivial idempotent are zero product determined. As applications of the above results, we prove that simple rings with a nontrivial idempotent, full matrix rings and some classes of operator algebras are zero product determined rings and discuss whether triangular rings and upper triangular matrix rings are zero product determined.