Let A be a unital algebra and M be a unital A-bimodule. A characterization of generalized derivations and generalized Jordan derivations from A into M, through zero products or zero Jordan products, is given. Suppose that M is a unital left A-module. It is investigated when a linear mapping from A into M is a Jordan left derivation under certain conditions. It is also studied whether an algebra...

Let $A$ be an algebra over a field $F$ with $(F)\ne 2$. If is generated as by $[[A,A],[A,A]]$, then for every skew-symmetric bilinear map $\Phi:A\times A\to X$, where $X$ arbitrary vector space $F$, the condition that $\Phi(x^2,x)=0 $ all $x\in A$ implies $\Phi(xy,z) +\Phi(zx,y) + \Phi(yz,x)=0$ $x,y,z\in A$. This applicable to question of whether zero Lie product determined and also used in pro...

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 de...