## 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 laterof the day

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 identity $1\neq 0$. Suppose that for each element $a\in R$, there exists an integer $n > 1$ depending on $a$.

Then prove that every prime ideal is a maximal ideal.

Add to solve laterGive an example of a commutative ring $R$ and a prime ideal $I$ of $R$ that is not a maximal ideal of $R$.

Add to solve later Let $R$ be a commutative ring. Consider the polynomial ring $R[x,y]$ in two variables $x, y$.

Let $(x)$ be the principal ideal of $R[x,y]$ generated by $x$.

Prove that $R[x, y]/(x)$ is isomorphic to $R[y]$ as a ring.

Add to solve later Let $R$ be a ring with unit $1$. Suppose that the order of $R$ is $|R|=p^2$ for some prime number $p$.

Then prove that $R$ is a commutative ring.

Let $R$ be a commutative ring with $1$ and let $M$ be an $R$-module.

Prove that the $R$-module $M$ is irreducible if and only if $M$ is isomorphic to $R/I$, where $I$ is a maximal ideal of $R$, as an $R$-module.

Let $R$ be a ring with $1$. Suppose that $R$ is an integral domain and an Artinian ring.

Prove that $R$ is a field.

**(a)** Let $R$ be an integral domain and let $M$ be a finitely generated torsion $R$-module.

Prove that the module $M$ has a nonzero annihilator.

In other words, show that there is a nonzero element $r\in R$ such that $rm=0$ for all $m\in M$.

Here $r$ does not depend on $m$.

**(b)** Find an example of an integral domain $R$ and a torsion $R$-module $M$ whose annihilator is the zero ideal.

Let $R$ be a commutative ring and let $I$ be a nilpotent ideal of $R$.

Let $M$ and $N$ be $R$-modules and let $\phi:M\to N$ be an $R$-module homomorphism.

Prove that if the induced homomorphism $\bar{\phi}: M/IM \to N/IN$ is surjective, then $\phi$ is surjective.

Add to solve later**(a)** Let $R$ be a commutative ring. If we regard $R$ as a left $R$-module, then prove that any two distinct elements of the module $R$ are linearly dependent.

**(b)** Let $f: M\to M’$ be a left $R$-module homomorphism. Let $\{x_1, \dots, x_n\}$ be a subset in $M$. Prove that if the set $\{f(x_1), \dots, f(x_n)\}$ is linearly independent, then the set $\{x_1, \dots, x_n\}$ is also linearly independent.

Read solution

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 and let $I_1$ and $I_2$ be **comaximal ideals**. That is, we have

\[I_1+I_2=R.\]

Then show that for any positive integers $m$ and $n$, the ideals $I_1^m$ and $I_2^n$ are comaximal.

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

Then show that every maximal ideal of $R$ is a prime ideal.

Let $R$ be the ring of all continuous functions on the interval $[0, 2]$.

Let $I$ be the subset of $R$ defined by

\[I:=\{ f(x) \in R \mid f(1)=0\}.\]

Then prove that $I$ is an ideal of the ring $R$.

Moreover, show that $I$ is maximal and determine $R/I$.

Let $R$ be a commutative ring with $1$ and let $G$ be a finite group with identity element $e$. Let $RG$ be the group ring. Then the map $\epsilon: RG \to R$ defined by

\[\epsilon(\sum_{i=1}^na_i g_i)=\sum_{i=1}^na_i,\]
where $a_i\in R$ and $G=\{g_i\}_{i=1}^n$, is a ring homomorphism, called the **augmentation map** and the kernel of $\epsilon$ is called the **augmentation ideal**.

**(a)** Prove that the augmentation ideal in the group ring $RG$ is generated by $\{g-e \mid g\in G\}$.

**(b)** Prove that if $G=\langle g\rangle$ is a finite cyclic group generated by $g$, then the augmentation ideal is generated by $g-e$.

Read solution

Let $R$ be a commutative ring with unity. A proper ideal $I$ of $R$ is called **primary** if whenever $ab \in I$ for $a, b\in R$, then either $a\in I$ or $b^n\in I$ for some positive integer $n$.

**(a)** Prove that a prime ideal $P$ of $R$ is primary.

**(b)** If $P$ is a prime ideal and $a^n\in P$ for some $a\in R$ and a positive integer $n$, then show that $a\in P$.

**(c)** If $P$ is a prime ideal, prove that $\sqrt{P}=P$.

**(d)** If $Q$ is a primary ideal, prove that the radical ideal $\sqrt{Q}$ is a prime ideal.

Let $R$ be an integral domain. Then prove that the ideal $(x^3-y^2)$ is a prime ideal in the ring $R[x, y]$.

Add to solve laterLet $R$ be a commutative ring with $1$. Show that if $R$ is an integral domain, then the characteristic of $R$ is either $0$ or a prime number $p$.

Add to solve laterConsider the ring

\[\Z[\sqrt{10}]=\{a+b\sqrt{10} \mid a, b \in \Z\}\]
and its ideal

\[P=(2, \sqrt{10})=\{a+b\sqrt{10} \mid a, b \in \Z, 2|a\}.\]
Show that $p$ is a prime ideal of the ring $\Z[\sqrt{10}]$.

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.

Add to solve later