Abstract Algebra: Ring Theory Problems

[Hotsuma]

Hello all,

I am trying to work on some Ring Theory proofs and my Abstract Algebra is very minimal as I have not taken the class but need to look into it nonetheless. If anyone can figure these out for me I'd greatly appreciate it.

Also, I am familiar in LaTeX typesetting but I don't know how to fluidly use it on this forum. If I enable the TeX can I use my typical math commands and implement conventional TeX'ing?

Homework Statement

This is the doozy for me:

If R1,R2,...,Rn are commutative rings with identity, shows that U (R1 x R2 x ... x Rn) = U(R1) x U(R2) x ... x U(Rn)

Homework Equations

Identity: There exists some 1 in R such that a*1 = 1*a = a
Commutativity: A, b in R such that ab = ba for all a, b.
Unit: Must have identity, there exist some y in R such that xy = yx =1. Must be commutative.

The Attempt at a Solution

I've tried all sorts of debauchery but nothing seems to work. I have tried splitting this up and looking at it different ways, but nothing I've come up with is correct.

Homework Statement

Show that a finite commutative ring with no zero-divisors and at least two elements has identity.

Homework Equations

See above.

The Attempt at a Solution

So I need to show it is finite (easy), commutative, no zero-divisors, and and at least two elements has identity.

I define some ring R with no zero-divisors and at least two elements. Let a,b be in R. There exists a*a=1; a*b= a (or b). If a*a=a, then a*a*b=a*b...

Here I just keep running in circles.

Homework Statement

Prove that (x) in Q[x] is maximal.

Homework Equations

Q is the set of all rationals.
http://en.wikipedia.org/wiki/Maximal_element"

The Attempt at a Solution

Nothing. =[

Thank you for all the help!

 

[HallsofIvy]

You use Latex on this board by enclosing it in [ tex ]...[ /tex ] or [ itex ]... [ /itex ] or [ latex] ... [ latex ] without the spaces.

$$\frac{a}{b}$$
$\frac{a}{b}$
$\frac{a}{b}$

As for your problem, what do you mean by U (R1 x R2 x ... x Rn) = U(R1) x U(R2) x ... x U(Rn)? What is that "U"?

 

[DorianG]

For your second problem, think of the definition of a ring. It is a group, with two operations (R,+,.) ... What do you know about the operation of addition in the ring? i.e. what must (R,+) be?