In
mathematics, a ring homomorphism is a structure-preserving
function between two
rings. More explicitly, if R and S are rings, then a ring homomorphism is a function that preserves addition, multiplication and
multiplicative identity; that is,[1][2][3][4][5]
for all in
These conditions imply that additive inverses and the additive identity are preserved too.
If in addition f is a
bijection, then its
inversef−1 is also a ring homomorphism. In this case, f is called a ring isomorphism, and the rings R and S are called isomorphic. From the standpoint of ring theory, isomorphic rings have exactly the same properties.
If R and S are
rngs, then the corresponding notion is that of a rng homomorphism,[a] defined as above except without the third condition f(1R) = 1S. A rng homomorphism between (unital) rings need not be a ring homomorphism.
The
composition of two ring homomorphisms is a ring homomorphism. It follows that the rings forms a
category with ring homomorphisms as
morphisms (see
Category of rings).
In particular, one obtains the notions of ring endomorphism, ring isomorphism, and ring automorphism.
Properties
Let f : R → S be a ring homomorphism. Then, directly from these definitions, one can deduce:
f(0R) = 0S.
f(−a) = −f(a) for all a in R.
For any
unita in R, f(a) is a unit element such that f(a)−1 = f(a−1) . In particular, f induces a
group homomorphism from the (multiplicative) group of units of R to the (multiplicative) group of units of S (or of im(f)).
The
kernel of f, defined as ker(f) = {a in R | f(a) = 0S}, is a
two-sided ideal in R. Every two-sided ideal in a ring R is the kernel of some ring homomorphism.
An homomorphism is injective if and only if kernel is the
zero ideal.
The
characteristic of Sdivides the characteristic of R. This can sometimes be used to show that between certain rings R and S, no ring homomorphism R → S exists.
If Rp is the smallest
subring contained in R and Sp is the smallest subring contained in S, then every ring homomorphism f : R → S induces a ring homomorphism fp : Rp → Sp.
If R is a
field (or more generally a
skew-field) and S is not the
zero ring, then f is injective.
If both R and S are
fields, then im(f) is a subfield of S, so S can be viewed as a
field extension of R.
If I is an ideal of S then f−1(I) is an ideal of R.
If R and S are commutative and P is a
prime ideal of S then f−1(P) is a prime ideal of R.
If R and S are commutative, M is a
maximal ideal of S, and f is surjective, then f−1(M) is a maximal ideal of R.
If R and S are commutative and S is an
integral domain, 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.
If f is surjective, P is prime (maximal) ideal in R and ker(f) ⊆ P, then f(P) is prime (maximal) ideal in S.
Moreover,
The composition of ring homomorphisms S → T and R → S is a ring homomorphism R → T.
For each ring R, the identity map R → R is a ring homomorphism.
Therefore, the class of all rings together with ring homomorphisms forms a category, the
category of rings.
The zero map R → S that sends every element of R to 0 is a ring homomorphism only if S is the
zero ring (the ring whose only element is zero).
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.
For every ring R, there is a unique ring homomorphism from R to the zero ring. This says that the zero ring is a
terminal object in the category of rings.
As the initial object is not isomorphic to the terminal object, there is no
zero object in the category of rings; in particular, the zero ring is not a zero object in the category of rings.
Examples
The function f : Z → Z/nZ, defined by f(a) = [an = a mod n is a
surjective ring homomorphism with kernel nZ (see
modular arithmetic).
The
complex conjugationC → C is a ring homomorphism (this is an example of a ring automorphism).
For a ring R of prime characteristic p, R → R, x → xp is a ring endomorphism called the
Frobenius endomorphism.
If R and S are rings, the zero function from R to S is a ring homomorphism if and only if S is the
zero ring (otherwise it fails to map 1R to 1S). On the other hand, the zero function is always a rng homomorphism.
If RX] denotes the ring of all
polynomials in the variable X with coefficients in the
real numbersR, and C denotes the
complex numbers, then the function f : RX] → 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 RX] that are divisible by X2 + 1.
If f : R → S is a ring homomorphism between the rings R and S, then f induces a ring homomorphism between the
matrix ringsMn(R) → Mn(S).
Let V be a vector space over a field k. Then the map ρ : k → End(V) given by ρ(a)v = av is a ring homomorphism. More generally, given an abelian group M, a module structure on M over a ring R is equivalent to giving a ring homomorphism R → End(M).
The function f : Z/6Z → Z/6Z defined by f([a6) = [4a6 is a rng homomorphism (and rng endomorphism), with kernel 3Z/6Z and image 2Z/6Z (which is isomorphic to Z/3Z).
There is no ring homomorphism Z/nZ → Z for any n ≥ 1.
If R and S are rings, the inclusion R → R × S that sends each r to (r,0) is a rng homomorphism, but not a ring homomorphism (if S is not the zero ring), since it does not map the multiplicative identity 1 of R to the multiplicative identity (1,1) of R × S.
A ring endomorphism is a ring homomorphism from a ring to itself.
A ring isomorphism is a ring homomorphism having a 2-sided inverse that is also a ring homomorphism. One can prove that a ring homomorphism is an isomorphism if and only if it is
bijective as a function on the underlying sets. If there exists a ring isomorphism between two rings R and S, then R and S are called isomorphic. Isomorphic rings differ only by a relabeling of elements. Example: Up to isomorphism, there are four rings of order 4. (This means that there are four pairwise non-isomorphic rings of order 4 such that every other ring of order 4 is isomorphic to one of them.) On the other hand, up to isomorphism, there are eleven
rngs of order 4.
A ring automorphism is a ring isomorphism from a ring to itself.
Monomorphisms and epimorphisms
Injective ring homomorphisms are identical to
monomorphisms in the category of rings: If f : R → S is a monomorphism that is not injective, then it sends some r1 and r2 to the same element of S. Consider the two maps g1 and g2 from Zx] to R that map x to r1 and r2, respectively; f ∘ g1 and f ∘ 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.
^Some authors use the term "ring" to refer to structures that do not require a multiplicative identity; instead of "rng", "ring", and "rng homomorphism", they use the terms "ring", "ring with identity", and "ring homomorphism", respectively. Because of this, some other authors, to avoid ambiguity, explicitly specify that rings are unital and that homomorphisms preserve the identity.
In
mathematics, a ring homomorphism is a structure-preserving
function between two
rings. More explicitly, if R and S are rings, then a ring homomorphism is a function that preserves addition, multiplication and
multiplicative identity; that is,[1][2][3][4][5]
for all in
These conditions imply that additive inverses and the additive identity are preserved too.
If in addition f is a
bijection, then its
inversef−1 is also a ring homomorphism. In this case, f is called a ring isomorphism, and the rings R and S are called isomorphic. From the standpoint of ring theory, isomorphic rings have exactly the same properties.
If R and S are
rngs, then the corresponding notion is that of a rng homomorphism,[a] defined as above except without the third condition f(1R) = 1S. A rng homomorphism between (unital) rings need not be a ring homomorphism.
The
composition of two ring homomorphisms is a ring homomorphism. It follows that the rings forms a
category with ring homomorphisms as
morphisms (see
Category of rings).
In particular, one obtains the notions of ring endomorphism, ring isomorphism, and ring automorphism.
Properties
Let f : R → S be a ring homomorphism. Then, directly from these definitions, one can deduce:
f(0R) = 0S.
f(−a) = −f(a) for all a in R.
For any
unita in R, f(a) is a unit element such that f(a)−1 = f(a−1) . In particular, f induces a
group homomorphism from the (multiplicative) group of units of R to the (multiplicative) group of units of S (or of im(f)).
The
kernel of f, defined as ker(f) = {a in R | f(a) = 0S}, is a
two-sided ideal in R. Every two-sided ideal in a ring R is the kernel of some ring homomorphism.
An homomorphism is injective if and only if kernel is the
zero ideal.
The
characteristic of Sdivides the characteristic of R. This can sometimes be used to show that between certain rings R and S, no ring homomorphism R → S exists.
If Rp is the smallest
subring contained in R and Sp is the smallest subring contained in S, then every ring homomorphism f : R → S induces a ring homomorphism fp : Rp → Sp.
If R is a
field (or more generally a
skew-field) and S is not the
zero ring, then f is injective.
If both R and S are
fields, then im(f) is a subfield of S, so S can be viewed as a
field extension of R.
If I is an ideal of S then f−1(I) is an ideal of R.
If R and S are commutative and P is a
prime ideal of S then f−1(P) is a prime ideal of R.
If R and S are commutative, M is a
maximal ideal of S, and f is surjective, then f−1(M) is a maximal ideal of R.
If R and S are commutative and S is an
integral domain, 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.
If f is surjective, P is prime (maximal) ideal in R and ker(f) ⊆ P, then f(P) is prime (maximal) ideal in S.
Moreover,
The composition of ring homomorphisms S → T and R → S is a ring homomorphism R → T.
For each ring R, the identity map R → R is a ring homomorphism.
Therefore, the class of all rings together with ring homomorphisms forms a category, the
category of rings.
The zero map R → S that sends every element of R to 0 is a ring homomorphism only if S is the
zero ring (the ring whose only element is zero).
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.
For every ring R, there is a unique ring homomorphism from R to the zero ring. This says that the zero ring is a
terminal object in the category of rings.
As the initial object is not isomorphic to the terminal object, there is no
zero object in the category of rings; in particular, the zero ring is not a zero object in the category of rings.
Examples
The function f : Z → Z/nZ, defined by f(a) = [an = a mod n is a
surjective ring homomorphism with kernel nZ (see
modular arithmetic).
The
complex conjugationC → C is a ring homomorphism (this is an example of a ring automorphism).
For a ring R of prime characteristic p, R → R, x → xp is a ring endomorphism called the
Frobenius endomorphism.
If R and S are rings, the zero function from R to S is a ring homomorphism if and only if S is the
zero ring (otherwise it fails to map 1R to 1S). On the other hand, the zero function is always a rng homomorphism.
If RX] denotes the ring of all
polynomials in the variable X with coefficients in the
real numbersR, and C denotes the
complex numbers, then the function f : RX] → 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 RX] that are divisible by X2 + 1.
If f : R → S is a ring homomorphism between the rings R and S, then f induces a ring homomorphism between the
matrix ringsMn(R) → Mn(S).
Let V be a vector space over a field k. Then the map ρ : k → End(V) given by ρ(a)v = av is a ring homomorphism. More generally, given an abelian group M, a module structure on M over a ring R is equivalent to giving a ring homomorphism R → End(M).
The function f : Z/6Z → Z/6Z defined by f([a6) = [4a6 is a rng homomorphism (and rng endomorphism), with kernel 3Z/6Z and image 2Z/6Z (which is isomorphic to Z/3Z).
There is no ring homomorphism Z/nZ → Z for any n ≥ 1.
If R and S are rings, the inclusion R → R × S that sends each r to (r,0) is a rng homomorphism, but not a ring homomorphism (if S is not the zero ring), since it does not map the multiplicative identity 1 of R to the multiplicative identity (1,1) of R × S.
A ring endomorphism is a ring homomorphism from a ring to itself.
A ring isomorphism is a ring homomorphism having a 2-sided inverse that is also a ring homomorphism. One can prove that a ring homomorphism is an isomorphism if and only if it is
bijective as a function on the underlying sets. If there exists a ring isomorphism between two rings R and S, then R and S are called isomorphic. Isomorphic rings differ only by a relabeling of elements. Example: Up to isomorphism, there are four rings of order 4. (This means that there are four pairwise non-isomorphic rings of order 4 such that every other ring of order 4 is isomorphic to one of them.) On the other hand, up to isomorphism, there are eleven
rngs of order 4.
A ring automorphism is a ring isomorphism from a ring to itself.
Monomorphisms and epimorphisms
Injective ring homomorphisms are identical to
monomorphisms in the category of rings: If f : R → S is a monomorphism that is not injective, then it sends some r1 and r2 to the same element of S. Consider the two maps g1 and g2 from Zx] to R that map x to r1 and r2, respectively; f ∘ g1 and f ∘ 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.
^Some authors use the term "ring" to refer to structures that do not require a multiplicative identity; instead of "rng", "ring", and "rng homomorphism", they use the terms "ring", "ring with identity", and "ring homomorphism", respectively. Because of this, some other authors, to avoid ambiguity, explicitly specify that rings are unital and that homomorphisms preserve the identity.