Find a concept
Start typing to search the mathematical index.
A subring modulo its intersection with an ideal is isomorphic to its image in the corresponding quotient.
Second isomorphism theorem (rings): Let be a ring, let be a subring, and let be a two-sided ideal. Then , is a subring of , and there is a natural isomorphism
of quotient rings.
This is most efficiently proved by restricting the quotient map to and applying the first isomorphism theorem.
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.