RING HOMOMORPHISM

In 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
(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.

Contents

Properties

Examples

Types of ring homomorphisms

See also

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'' ∈ ''R'' : ''f''(''a'') = 0} is an ideal in ''R''. Every ideal in a commutative ring ''R'' arises from some ring homomorphism in this way, but this is never true for a non-commutative ring. ''f'' is injective if and only if the ker(''f'') = {0}. Note that in general, for rings with identity the kernel of a ring homomorphism is not a subring since it will not contain the multiplicative identity.

★ 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 ''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''. This can sometimes be used to show that between certain rings ''R'' and ''S'', no ring homomorphisms ''R'' → ''S'' can exist.

★ 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''.

★ 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 ''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.

See also

★ homomorphism

★ ring theory