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

[curiousmuch]

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.

 

[Dick]

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.

 

[Dick]

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.

 

[Dick]

http://en.wikipedia.org/wiki/Ring_(mathematics)#Formal_definition See 'closure axiom for multiplication'.

 

[Dick]

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.