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The kernel of a ring homomorphism is a two-sided ideal of the domain.
Kernel is an ideal: Let be a ring homomorphism. Then
is a two-sided ideal of .
For a ring homomorphism , the kernel is therefore an ideal (indeed a two-sided ideal in general), so one can form the quotient ring . This is the key input for the First Isomorphism Theorem for rings.
An ideal in a ring is an additive subgroup such that it is stable under multiplication by elements of on one side:
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.