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The definition of homomorphism depends on the type of algebraic structure under consideration. The common theme is that a homomorphism is a function between two algebraic objects that respects the algebraic structure. — “Homomorphism - Wikipedia, the free encyclopedia”, en.wikipedia.org
In mathematics, given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G H such that for all u and v in G it holds that. where the group operation on the left hand side of the equation is that of G and on the right hand side that of H. — “Group homomorphism - Wikipedia, the free encyclopedia”, en.wikipedia.org
In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication. More precisely, if R and S are rings, then a ring homomorphism is a function f : R S such that f(a + b) = f(a) + f(b) for all a and b in R. — “Ring homomorphism - Wikipedia, the free encyclopedia”, en.wikipedia.org