We have seen generation as a means of constructing ideals. Here we discuss two more ways of obtaining ideals. Let R be a commutative ring.
For subsets X
and Y of R, the sum X + Y
is the subset {a + b | a
X , b Y} of R.
Proposition
If I and J are ideals of R, then the sum I + J is an ideal of R.
Let S also be a commutative ring.
The kernel of a morphism
f : R -> S is the subset
{x R | f(x) = 0} of R.
Theorem
If f : R -> S is a morphism of rings, then the kernel Ker(f) is an ideal of R.
We shall see later that every proper ideal of R can be seen as the kernel
of some morphism.