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A ring homomorphism induces an isomorphism from the quotient by its kernel onto its image.
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.
This identifies the universal quotient quotient ring determined by with the concrete subring realized as its image, and is the basic tool behind “modding out by relations” in ring constructions.
A ring isomorphism is a ring homomorphism that is bijective and whose inverse function is also a ring homomorphism.
The image of a ring homomorphism is
i.e. the image of under .
The image is a subring of (and a unital subring if is unital and is unital). Together with the kernel, the image governs the structure of via isomorphism theorems.
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.