Answer to Question #178768 in Abstract Algebra for Zainab

Question #178768

Suppose that R is a ring with identity such that char R=n>0 .if n is not prime, show that char R=n>0.if n is not prime, show that R has divisors of zero

Expert's answer

Let's not assume R has a unit and is not the zero ring. Consider

If , then obviously . If and , with and , then

and rs≠0. Therefore m+n∈I and we have proved that I is an ideal of Z, because obviously , as R≠{0}. Also

Thus for a unique . If with , then ar≠0 and br≠0 for any Let r≠0 with : then ar≠0 and br≠0 because but

which is a contradiction. Thus either k=0 or k is a prime.

In the first case nr≠0 for every , and every , so that every nonzero element of R has infinite order.

Suppose k is prime; we want to show that , for every . Assume the contrary and let

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