Zero-product balanced algebras

We say that an algebra is zero-product balanced if ab⊗c and a⊗bc agree modulo tensors of elements with zero-product. This closely related to but more general than the notion a determined introduced developed by Brešar, Villena others. Every surjective, preserving map from automatically weighted epimorphism, this implies algebras are their linear structure. Further, commutator subspace can be described in terms square-zero elements. show commutative, reduced only it generated idempotents. It follows every spanned nilpotent idempotent