Find a concept
Start typing to search the mathematical index.
The kernel of a ring homomorphism is always a two-sided ideal.
Kernels are two-sided ideals: If is a ring homomorphism, then its kernel is a two-sided ideal of .
This lemma is a prerequisite for forming the associated quotient ring and is used immediately in the first isomorphism theorem for rings.
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.