## Every Ring of Order $p^2$ is Commutative

## Problem 501

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 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 ring with $1$. Prove that the following three statements are equivalent.

- The ring $R$ is a field.
- The only ideals of $R$ are $(0)$ and $R$.
- Let $S$ be any ring with $1$. Then any ring homomorphism $f:R \to S$ is injective.

Let $\Z$ be the ring of integers and let $R$ be a ring with unity.

Determine all the ring homomorphisms from $\Z$ to $R$.

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.