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A ring-homomorphism between rings
and
is a mapping
such that
is a monoid-homomorphism for the multiplicative structures on
and
, and simultaneously also a monoid-homomorphism for their additive structures. In other words,
satisfies
for all , where
, and
denote the identity and zero element of
, and
, respectively.
Any ring-homomorphism which is one-to-one is called an isomorphism.
If is an ideal of the ring
, and
the corresponding factor ring (also called a residue class ring), then the canonical map
is a surjective ring-homomorphism, called the natural quotient map or the canonical homomorphism..
Theorem: Let be a ring-homomorphism. Then the image
of
is a subring of
.
An injective (one-to-one) ring-homomorphism establishes a ring-isomorphism between
and its image. Such a homomorphism is called an embedding (of a ring
into
).
The kernel of a ring-homomorphism
is an ideal of
.
Theorem: If is a ring-homomorphism whose kernel contains
, and
is the canonical homomorphism, then there exists a unique ring-homomorphism
making the following diagram commutative
The last theorem can be equivalently rephrased saying that the canonical map is universal in the category of homomorphisms whose kernel contains the ideal
.
Cite this web-page as:
Štefan Porubský: Ring homomorphism.