A module homomorphism is a map f:M->N between modules over a ring R which preserves both the addition and the multiplication by scalars. In symbols this means that f(x + y) = f(x) + f(y) for all x, y element M and f(a x) = a f(x) for all x, element M, for all a element R. Note that if the ring R is replaced by a field K, these conditions yield exactly the definition of f as a linear map between abstract vector spaces over K. For all modules M over a commutative ring R, and all a element R, the multiplication by a determines a module homomorphism μ_a :M->M, defined by μ_a(x) = a x for all x element M.
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