As of May 4 2007 the scripts will autodetect your timezone settings. Nothing here has to be changed, but there are a few things

## Tuesday, July 17, 2007

### 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.

## Welcome to The Bridge

Mathematics: is it the fabric of MEST?
This is my voyage
My continuous mission
To uncover hidden structures
To create new theorems and proofs
To boldly go where no man has gone before

(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)