Homomorphisms are the maps between algebraic objects. There are two main types: group homomorphisms and ring homomorphisms. (Other examples include vector spacehomomorphisms, which are generally called linear maps, as well as homomorphisms of modules and homomorphisms of algebras.)
Generally speaking, a homomorphism between two algebraic objects A, B is a function f : A → B which preserves the algebraic structure on A and B That is, if elements in A satisfy some algebraic equation involving addition or multiplication, their images in B satisfy the same algebraic equation. The details of the definitions of homomorphisms in various contexts depend on the algebraic structures of A and B.
If the operations on A and B are both addition, then the homomorphism condition is If A and B are both rings, with addition and multiplication, there is also a multiplicative condition:
A bijective homomorphism is called an isomorphism. An isomorphism between two algebraic objects and identifies them with each other; they are, in an algebraic sense, the same object (possibly written in two different ways). The most common use of homomorphisms in abstract algebra is via the three so-called isomorphism theorems, which allow for the identification of certain quotient objects with certain other subobjects (subgroups, subrings, etc.)
The study of the interplay between algebraic objects is fundamental in the study of algebra. The existence and properties of homomorphisms from one algebraic object to another give a rich depth of information about the objects and their relationship. Many important concepts in abstract algebra, such as
the integers modulo n
a prime ideal in a ring
the sign of a permutation,
can be naturally considered as (respectively) the image of a homomorphism, the kernel of a homomorphism, or the homomorphism itself.
Definitions and Examples
|Let A and B be groups, with operations given by ºA and ºB respectively. A group homomorphism f : A → B is a function f such that for all x,y ∈ A.|
Let R and S be rings, with operations + and . (this is a slight abuse of notation, but the formulas below are more unwieldy with subscripts on the operations). A ring homomorphism f : R → S is a function f such that
(In this wiki, "ring" means "ring with unity"; a homomorphism of rings is defined in the same way, but without the third condition.)
In both cases, a homomorphism is called an isomorphism if it is bijective.
Show that if f : R → S is a ring homomorphism, f(0R) = os.
Note that by the homomorphism property. Since f(0R) has an additive inverse in S, we can add it to both sides of this equation to get 0S = f(0R).
1. For any groups G and H, there is a trivial homomorphis 2. Let be a positive integer. The function is a ring homomorphism (and as such, it is a homomorphism of additive groups).3. Define is complex conjugation. Then c is a homomorphism from C to itself. It is clearly a bijection, so it is in fact an isomorphism from C to itself.4. Let R be a subring of S, and pick Then there is an evaluation homomorphism is the ring of polynomials with coefficients in R.It is given by 5. The map is a group homomorphism. Note that R is an additive group and R* the set of nonzero real numbers, is a multiplicative group. The verification that f is a group homomorphism is precisely the law of exponents: 6. Let Sn be the symmetric group on n letters. There is a unique nontrivial group homomorphism the latter being a group under multiplication. The value is called the sign of σ, and is important in many applications, including one definition of the determinant of a matrix.
Kernel and Image
Any homomorphism f : A → B has two objects associated to it: the kernel, which is a subset of A, and the image, which is a subset of B.
be a group homomorphism. The kernel of f, ker (f), is the subset of G consisting of elements G such that is the group identity element).
be a ring homomorphism. The kernel of is the subset of R consisting of elements R such that
For further exploration of the kernel in the setting of vector spaces, see the wiki.
The kernel of a homomorphism is an important object, in both group and ring theory. The following theorem identifies what kind of object it is:
Continuing the six examples above:
1. If is the trivial homomorphism, then ker the trivial subgroup of H2. The kernel of reduction mod n is the ideal consisting of multiples of n. The image is all of Zn; reduction mod n is surjective.
Properties of Homomorphisms