Re: Finite Commutative Ring

> Please help with the following practice prelim
> question:
>
> Show that if R is a finite commutative ring such that
> |R| = p, a prime, then R is a field.
>
> Thanks in advance.
>
> --Jim


Obviously, your defintion of a "ring" includes the existence of an identity element 1, otherwise the zeroring of order p would be a counterexample. Since R has the prime order p, every nonzero element is a generator of the additive group. If we denote for any a with 0<a<p the element

1+1...+1 (a times)

by a, then due to the distributivity laws the multiplication for any two elements x,y in R is given by

x*y = xy mod p

Hence, this ring is the residue class ring mod p, whioh is known to be a field.

Johann
.

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