Ideals

Let be a ring. A subset is an ideal in if

1. is an additive subgroup of , and
2. for every and .
   If is an ideal of , we write .

1. for any .
2. for any .
   These are the trivial ideals. Note that if is a field, then these are the only ideals since every element can be reached from any element (except ) by multiplication. This also means has no non-trivial ideals.
3. . These are all the ideals of !

Let be a commutative ring.

1. Let . Then is the smallest ideal in containing . This is called the principal ideal generated by .
2. Let . Then is the smallest ideal in containing . If , we say that is finitely generated.

Let be a ring, . Then

1. .
2. .
3. .