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The kernel of a ring homomorphism is a two-sided ideal of the domain.
Let be a ring homomorphism. Then
is a two-sided ideal of .
One can therefore form the quotient ring . This is the key input for the first isomorphism theorem for rings.
Let be a ring and an additive subgroup. Then is a left ideal if for every , a right ideal if for every , and a two-sided ideal if both conditions hold.
Let be a ring and let be a two-sided ideal of . The quotient ring is the quotient set of by the equivalence relation iff , with addition and multiplication on cosets defined by
The two-sided condition ensures multiplication is well-defined.
The canonical projection is a surjective ring homomorphism with kernel , and satisfies the universal property of quotients.
First isomorphism theorem (rings): Let be a ring homomorphism. Then the induced map
is a ring isomorphism, where is the kernel and is the image.