Let R and R' be rings.
Definition
A morphism f : R -> R' is a map from R to R' such that f is
- a morphism of additive groups f : R -> R', and
- a morphism of the multiplicative monoid of R to that of R'.
Let f : R -> R' be a morphism.
- The kernel of f is the
set {a
R | f(a) =
0}; it is denoted by Ker(f).
- The image of
f is the set f(R) = {f(a) |
a R}; it is denoted by Im(f).
So a map f : R -> R' is a morphism if the following conditions are satisfied.
- f(0) = 0;
- f(a + b) = f(a) + f(b);
- f(1) = 1;
- f(a * b) = f(a) * f(b).
A bijective morphism is called an isomorphism.
Theorem
For every morphism f : R -> R' the following holds.
- If a R is invertible, then
so is f(a); its inverse is
f(a)-1 = f(a-1).
- If f is an isomorphism, then the cardinalities of R and R' are equal.
- If f is an isomorphism, then so is its inverse f-1 : R' -> R.
- f(R) is a subring of R'.
- Ker(f) is an additive subgroup
of R; if a is in Ker(f), then ra is in
Ker(f) for all r R.
- f is injective if and only if Ker(f) = {0}.