Question

If R is a finite commutative ring with identity, prove that every prime ideal in R is maximal. [Hint: Theorem 3.9.]

Short answer

It is proved that every prime ideal is maximal.

Step-by-step solution

Obtain that R/I is an integral domain

Consider that R is a finite commutative ring with identity.

Here, I is a prime ideal in R .

Using the theorem 6.14, we can say that R/I is an integral domain.

Obtain that every prime ideal is maximal

As R is finite then R/I is also finite.

By theorem 3.9, it is clear that R/I is filed.

And by theorem 6.15, that is, Consider P be an ideal in a commutative ring R with the identity. This implies that P is a prime ideal if and only if the quotient ring R / P is an integral domain. This implies that the ideal I is maximal.