  We reformulate the definition of a zero product determined algebra in terms
of tensor products and obtain necessary and sufficient conditions for an
algebra to be zero product determined. These conditions allow us to prove that
the direct sum \bigoplus_{i \in I} A_i of algebras for any index set I is zero
product determined if and only if each of the component algebras A_i is zero
product determined. As an application, every parabolic subalgebra of a
finite-dimensional reductive Lie algebra, over an algebraically-closed field of
characteristic zero, is zero product determined. In particular, every such
reductive Lie algebra is zero product determined.
