Ring homomorphism explained

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

f:R \to S that preserves addition, multiplication and multiplicative identity; that is,\begin{align} f(a + b)& = f(a) + f(b),\\ f(ab) & = f(a)f(b), \\ f( 1 _{R)} & = 1 _{S, \end{align}} for all a,b in R.

These conditions imply that additive inverses and the additive identity are preserved too.

If in addition is a bijection, then its inverse ⁻¹ is also a ring homomorphism. In this case, 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, defined as above except without the third condition f(1_R) = 1_S. 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 be a ring homomorphism. Then, directly from these definitions, one can deduce:

• f(0_R) = 0_S.
• f(−a) = −f(a) for all a in R.
• For any unit a in R, f(a) is a unit element such that . 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 image of f, denoted im(f), is a subring of S.
• The kernel of f, defined as, 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 S divides the characteristic of R. This can sometimes be used to show that between certain rings R and S, no ring homomorphism exists.
• If R_p is the smallest subring contained in R and S_p is the smallest subring contained in S, then every ring homomorphism induces a ring homomorphism .
• If R is a field (or more generally a skew-field) and S is not the zero ring, then 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 ⁻¹(I) is an ideal of R.
• If R and S are commutative and P is a prime ideal of S then ⁻¹(P) is a prime ideal of R.
• If R and S are commutative, M is a maximal ideal of S, and is surjective, then ⁻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 is surjective, then ker(f) is a maximal ideal of R.
• If is surjective, P is prime (maximal) ideal in R and, then f(P) is prime (maximal) ideal in S.

Moreover,

• The composition of ring homomorphisms and is a ring homomorphism .
• For each ring R, the identity map is a ring homomorphism.
• Therefore, the class of all rings together with ring homomorphisms forms a category, the category of rings.
• The zero map 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 . 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, defined by is a surjective ring homomorphism with kernel nZ (see modular arithmetic).
• The complex conjugation is a ring homomorphism (this is an example of a ring automorphism).
• For a ring R of prime characteristic p, 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 1_R to 1_S). On the other hand, the zero function is always a rng homomorphism.
• 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 defined by (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''] that are divisible by .
• If is a ring homomorphism between the rings R and S, then f induces a ring homomorphism between the matrix rings .
• Let V be a vector space over a field k. Then the map given by 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 .
• A unital algebra homomorphism between unital associative algebras over a commutative ring R is a ring homomorphism that is also R-linear.

Non-examples

• The function defined by 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 for any .
• If R and S are rings, the inclusion 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 .

Category of rings

See main article: Category of rings.

Endomorphisms, isomorphisms, and automorphisms

• 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 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 Z[''x''] to R that map x to r₁ and r₂, respectively; and are identical, but since is a monomorphism this is impossible.

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

See also

• Change of rings

References

• Book: Artin, Michael . Michael Artin . 1991 . Algebra . Englewood Cliffs, N.J. . Prentice Hall.
  • Book: Bourbaki . N. . Nicolas Bourbaki . 1998 . Algebra I, Chapters 1–3 . Springer.
• Book: Eisenbud, David . David Eisenbud . 1995 . Commutative algebra with a view toward algebraic geometry . New York . . . 150 . 1322960 . 0-387-94268-8 . true . xvi+785.
• Book: Hazewinkel . Michiel . Michiel Hazewinkel . 2004 . Algebras, rings and modules . . 1-4020-2690-0.
• Book: Jacobson . Nathan . Nathan Jacobson . 1985 . Basic algebra I . 2nd . 9780486471891 .