Homework 3 Solutions.

x5.3, #8 Show that if Ris a finite ring, then every prime ideal of Ris maximal. Proof. Let Rbe a finite ring and let P/Rbe a prime ideal. Since Pis prime, the quotient R=Pis an integral domain; since Ris finite, the quotient is finite. Hence, R=Pis a finite integral domain. As such, it is a field, and therefore, Pis maximal. ................
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