## If Every Proper Ideal of a Commutative Ring is a Prime Ideal, then It is a Field.

## Problem 598

Let $R$ be a commutative ring with $1$.

Prove that if every proper ideal of $R$ is a prime ideal, then $R$ is a field.

Add to solve laterLet $R$ be a commutative ring with $1$.

Prove that if every proper ideal of $R$ is a prime ideal, then $R$ is a field.

Add to solve later Let $R$ be an integral domain and let $I$ be an ideal of $R$.

Is the quotient ring $R/I$ an integral domain?

Let $R$ be a ring with $1\neq 0$. Let $a, b\in R$ such that $ab=1$.

**(a)** Prove that if $a$ is not a zero divisor, then $ba=1$.

**(b)** Prove that if $b$ is not a zero divisor, then $ba=1$.

**(a)** Let $F$ be a field. Show that $F$ does not have a nonzero zero divisor.

**(b)** Let $R$ and $S$ be nonzero rings with identities.

Prove that the direct product $R\times S$ cannot be a field.

Let $R$ be a ring with $1$. An element of the $R$-module $M$ is called a **torsion element** if $rm=0$ for some nonzero element $r\in R$.

The set of torsion elements is denoted

\[\Tor(M)=\{m \in M \mid rm=0 \text{ for some nonzero} r\in R\}.\]

**(a)** Prove that if $R$ is an integral domain, then $\Tor(M)$ is a submodule of $M$.

(Remark: an integral domain is a commutative ring by definition.) In this case the submodule $\Tor(M)$ is called **torsion submodule** of $M$.

**(b)** Find an example of a ring $R$ and an $R$-module $M$ such that $\Tor(M)$ is not a submodule.

**(c)** If $R$ has nonzero zero divisors, then show that every nonzero $R$-module has nonzero torsion element.

Let $R$ be a commutative ring. Suppose that $P$ is a prime ideal of $R$ containing no nonzero zero divisor. Then show that the ring $R$ is an integral domain.

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