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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication.A bijective ring homomorphism is called a ring isomorphism. A ring homomorphism whose domain is the same as its range is called a ring endomorphism. A ring automorphism is a bijective endomorphism.Injective ring homomorphisms are identical to monomorphisms in the category of rings: If f:R S is a monomorphism which is not injective, then it sends some r1 and r2 to the same element of S. Consider the two maps g1 and g2 from Z[x] to R which map x to r1 and r2, respectively; f o g1 and f o g2 are identical, but since f is a monomorphism this is impossible.