Ideals of a field.

I have found a page with a lot of algebra lecture notes of which a few in Dutch. To the point: some new insights I gained today.


Lemma 1:
Let R be a ring and I be an ideal of R. I=R IFF I contains a unit.

Proof:
Assume u is a unit in I. Then R contains an element v such that uv=1. Since I is an ideal uv is in I, or 1 in I. Thus I = <1> = R. Conversely if I = R then I contains the unit 1.


Lemma 2:
A commutative ring R is a field IFF its ideals are {0} and R.

Proof:
Assume I is an ideal of a field R with element u. Since R is a field there is an element v such that uv = 1. Since I is an ideal 1 is in I or I = <1> = R. Conversely, let R have ideals {0} and R. Assume R = (u) and thus 1 in (u) according to Lemma 1. So R has some v such that uv = 1, or every nonzero element in R is a unit and thus R is a field.


Let R,S be rings and f a nonzero ring homomorphism f: R->S. If R is a field then f is injective.

Proof:
According to lemma 2 R has two ideals {0} and R. The ideals are the kernel of a homomorphism. If the kernel of a ring homomorphism is R then it is the zero homomorphism. A homomorphism with kernel {0} is an isomorphism which is injective.