Finite Commutative Ring: Proving Integral Domain w/ No Zero Divisors

  • Thread starter Thread starter curiousmuch
  • Start date Start date
  • Tags Tags
    Rings
curiousmuch
Messages
7
Reaction score
0

Homework Statement


Show that a finite commutative ring with no zero divisors is an integral domain (i.e. contains a unity element)


Homework Equations


If a,b are elements in a ring R, then ab=0 if and only if either a and b are 0.


The Attempt at a Solution


I've been trying to use the cancellation laws.
 
Physics news on Phys.org
Let N be the set of all nonzero elements of the ring R. Pick a nonzero element c. Can you show cN=N? Remember N is a finite set.
 
curiousmuch said:
thanks, but we can't assume R is closed under multiplication.

What do you mean? R is a ring. A 'ring' is closed under its multiplication operation.
 
The clue is in post 2. Can you show multiplication by any nonzero element c maps the set of nonzero elements of the ring to itself in a one-to-one manner.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
View full post »

Thread 'Finding the nth roots of a complex number'

There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
View full post »

Similar threads

Nontrivial Finite Rings with No Zero Divisors
Replies
3
Views
3K
Abstract Algebra: Ring Proof (Multiplicative Inverse)
Replies
1
Views
2K
Can Zero Divisors in Polynomial Rings Be Characterized?
Replies
13
Views
2K
Invertible elements in a commutative ring with no zero divisors
Replies
3
Views
3K
Units / Zero divisors in Comm Ring
Replies
24
Views
2K
Showing that a ring is an integral domain
Replies
10
Views
2K
Proving a Product of Commutative Rings not an Integral Domain:
Replies
2
Views
3K
Abstract Algebra: Non-trivial Rings Containing Only Zero-Divisors
Replies
7
Views
3K
I Understanding zero divisors & ##\mathbb{Z_m}## in Ring Theory
2
Replies
55
Views
6K
Factor Ring of a Ring: Example of Integral Domain with Divisors of 0
Replies
3
Views
2K

Hot Threads

Recent Insights

Back
Top