Ring homomorphism

In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication.

More precisely, if "R" and "S" are rings, then a ring homomorphism is a function "f" : "R" → "S" such that
* "f"("a" + "b") = "f"("a") + "f"("b") for all "a" and "b" in "R"
* "f"("ab") = "f"("a") "f"("b") for all "a" and "b" in "R"
* "f"(1) = 1

Naturally, if one does not require rings to have a multiplicative identity then the last condition is dropped.

The composition of two ring homomorphisms is a ring homomorphism. It follows that the class of all rings forms a category with ring homomorphisms as the morphisms (cf. the category of rings).

Properties

Directly from these definitions, one can deduce:
* "f"(0) = 0
* "f"(−"a") = −"f"("a")
* If "a" has a multiplicative inverse in "R", then "f"("a") has a multiplicative inverse in "S" and we have "f"("a"⁻¹) = ("f"("a"))⁻¹. Therefore, "f" induces a group homomorphism from the group of units of "R" to the group of units of "S".
* The kernel of "f", defined as ker("f") = {"a" in "R" : "f"("a") = 0} is an ideal in "R". Every ideal in a commutative ring "R" arises from some ring homomorphism in this way. For rings with identity the kernel of a ring homomorphism is a subring without identity.
* The homomorphism "f" is injective if and only if the ker("f") = {0}.
* The image of "f", im("f"), is a subring of "S".
* If "f" is bijective, then its inverse "f"⁻¹ is also a ring homomorphism. "f" is called an isomorphism in this case, and the rings "R" and "S" are called isomorphic. From the standpoint of ring theory, isomorphic rings cannot be distinguished.
* If there exists a ring homomorphism "f" : "R" → "S" then the characteristic of "S" divides the characteristic of "R". This can sometimes be used to show that between certain rings "R" and "S", no ring homomorphisms "R" → "S" can exist.
* If "R_p" is the smallest subring contained in "R" and "S_p" is the smallest subring contained in "S", then every ring homomorphism "f" : "R" → "S" induces a ring homomorphism "f_p" : "R_p" → "S_p".
* If "R" is a field, then "f" is either injective or "f" is the zero function. (Note, however, that if "f" preserves the multiplicative identity, then it cannot be the zero function.)
* If both "R" and "S" are fields, then im("f") is a subfield of "S" (if "f" is not the zero function).
* If "R" and "S" are commutative and "S" has no zero divisors, then ker("f") is a prime ideal of "R".
* If "R" and "S" are commutative, "S" is a field, and "f" is surjective, then ker("f") is a maximal ideal of "R".
* For every ring "R", there is a unique ring homomorphism Z → "R". This says that the ring of integers is an initial object in the category of rings.

Examples

* The function "f" : Z → Z_"n", defined by "f"("a") = ["a"] _"n" = "a" mod "n" is a surjective ring homomorphism with kernel "n"Z (see modular arithmetic).
* There is no ring homomorphism Z_"n" → Z for "n" > 1.
* If R ["X"] denotes the ring of all polynomials in the variable "X" with coefficients in the real numbers R, and C denotes the complex numbers, then the function "f" : R ["X"] → C defined by "f"("p") = "p"("i") (substitute the imaginary unit "i" for the variable "X" in the polynomial "p") is a surjective ring homomorphism. The kernel of "f" consists of all polynomials in R ["X"] which are divisible by "X"² + 1.
* If "f" : "R" → "S" is a ring homomorphism between the "commutative" rings "R" and "S", then "f" induces a ring homomorphism between the matrix rings M_"n"("R") → M_"n"("S").

Types of ring homomorphisms

* A bijective ring homomorphism is called a "ring isomorphism".
* A ring homomorphism whose domain is the same as its range is called a "ring endomorphism".

Injective ring homomorphisms are identical to monomorphisms in the category of rings: If "f":"R"→"S" is a monomorphism which is not injective, then it sends some "r₁" and "r₂" to the same element of "S". Consider the two maps "g₁" and "g₂" from Z ["x"] to "R" which map "x" to "r₁" and "r₂", respectively; "f" o "g₁" and "f" o "g₂" are identical, but since "f" is a monomorphism this is impossible.

However, surjective ring homomorphisms are vastly different from epimorphisms in the category of rings. For example, the inclusion Z ⊆ Q is a ring epimorphism, but not a surjection. However, they are exactly the same as the strong epimorphisms.

ee also

*homomorphism