On zero product determined algebras

Abstract

Let be a commutative ring with identity. A -algebra is said to be zero product determined if for every -bilinear having the property that whenever , there is a -linear such that for all . We provide a necessary and sufficient condition for an algebra to be zero product determined and use the condition to derive several new results. Among these, we show that the direct sum of algebras is zero product determined if and only if each component algebra is zero product determined; we show that the tensor product of zero product determined algebras is zero product determined in case is a field or in case the algebra multiplications are surjective; we produce conditions under which the homomorphic images of a zero product determined algebra are zero product determined; finally, we introduce a class of zero product determined matrix algebras that generalizes block upper triangular matrices and extends a result of Brešar, Grašič, and Ortega in 2009.