For objects A and B belonging to a category C we denote by C(A,B) the set of all morphisms from A to B in C and, for any f ∈ C(A,B), the objects A and B are called, respectively, the domain and codomain of f . For example, if A and B are abelian groups, and we denote by U(A) and U(B) the underlying sets of A and B, then Ab(A,B) is the set of all group homomorphisms having domain A and codomain ...

martindale proved that under some conditions every multiplicative isomorphism between two rings is additive. in this paper, we extend this theorem to a larger class of mappings and conclude that every multiplicative $(alpha, beta)-$derivation is additive.

Graph homomorphism has been studied intensively. Given an m ×m symmetric matrix A, the graph homomorphism function is defined as

With a graph G = (V,E) we associate a collection of non-negative real weights ⋃ v∈V {λi,v : 1 ≤ i ≤ m} ∪ ⋃ uv∈E{λij,uv : 1 ≤ i ≤ j ≤ m}. We consider the probability distribution on {f : V → {1, . . . ,m}} in which each f occurs with probability proportional to ∏ v∈V λf(v),v ∏ uv∈E λf(u)f(v),uv . Many well-known statistical physics models, including the Ising model with an external field and the...