## Is the Quotient Ring of an Integral Domain still an Integral Domain?

## Problem 589

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 an integral domain and let $I$ be an ideal of $R$.

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

Let $I$ be a nonzero ideal of the ring of Gaussian integers $\Z[i]$.

Prove that the quotient ring $\Z[i]/I$ is finite.

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

\[P=(2, \sqrt{10})=\{a+b\sqrt{10} \mid a, b \in \Z, 2|a\}\]
be an ideal of the ring

\[\Z[\sqrt{10}]=\{a+b\sqrt{10} \mid a, b \in \Z\}.\]
Then determine the quotient ring $\Z[\sqrt{10}]/P$.

Is $P$ a prime ideal? Is $P$ a maximal ideal?

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 $R$ 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 $R/I$.