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## #1 2010-04-16 05:30:38

GOKILL
Member
Registered: 2010-03-19
Posts: 26

### Abstract Algebra - Ring !!!!!

#1
Let F be a field, show that F[x] is the main ideal domain (ideal region)!

#2
Let R be integral domain,

be prime ideal, and S = R\P.
a) Show that S doesn't contain zero-divisor
b) Defined
, show that Rs isomorfic with the subring of Q(R)
[Q(R) is the smallest subfield containing R, called as divisor field]

#3
Suppose f is ring homomorphism from R (ring with unity element) to the ring R'.
If

is unit, show that
.

thxu

I am the greatest magician this century!!!

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## #2 2010-04-16 10:10:49

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

### Re: Abstract Algebra - Ring !!!!!

1.  If you know the division algorithm works in F[x], then this proof becomes easy.  Prove it's a Euclidean domain, and this implies it is a principal ideal domain.

2b "show that Rs isomorfic with the subring of Q(R)" With a subring of Q(R)?  When you have a subset, there is always a very easy way to injectively define a map.

3 f(ab) = f(a)f(b)

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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