## 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 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 $\Z[x]$ be the ring of polynomials with integer coefficients.

Prove that

\[I=\{f(x)\in \Z[x] \mid f(-2)=0\}\]
is a prime ideal of $\Z[x]$. Is $I$ a maximal ideal of $\Z[x]$?

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

Suppose that $a, b$ are elements in $R$ such that

\[ab=1 \text{ and } ba\neq 1.\]

**(a)** Prove that $1-ba$ is idempotent.

**(b)** Prove that $b^n(1-ba)$ is nilpotent for each positive integer $n$.

**(c)** Prove that the ring $R$ has infinitely many nilpotent elements.

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 $ba=1$.

Let $R$ and $S$ be rings with $1\neq 0$.

Prove that every ideal of the direct product $R\times S$ is of the form $I\times J$, where $I$ is an ideal of $R$, and $J$ is an ideal of $S$.

Add to solve later**(a)** Prove that every prime ideal of a Principal Ideal Domain (PID) is a maximal ideal.

**(b)** Prove that a quotient ring of a PID by a prime ideal is a PID.

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 laterLet $R$ and $S$ be rings. Suppose that $f: R \to S$ is a surjective ring homomorphism.

Prove that every image of an ideal of $R$ under $f$ is an ideal of $S$.

Namely, prove that if $I$ is an ideal of $R$, then $J=f(I)$ is an ideal of $S$.

**(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 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 later A ring is called **local** if it has a unique maximal ideal.

**(a)** Prove that a ring $R$ with $1$ is local if and only if the set of non-unit elements of $R$ is an ideal of $R$.

**(b)** Let $R$ be a ring with $1$ and suppose that $M$ is a maximal ideal of $R$.

Prove that if every element of $1+M$ is a unit, then $R$ is a local ring.

Let

\[R=\left\{\, \begin{bmatrix}

a & b\\

0& a

\end{bmatrix} \quad \middle | \quad a, b\in \Q \,\right\}.\]
Then the usual matrix addition and multiplication make $R$ an ring.

Let

\[J=\left\{\, \begin{bmatrix}

0 & b\\

0& 0

\end{bmatrix} \quad \middle | \quad b \in \Q \,\right\}\]
be a subset of the ring $R$.

**(a)** Prove that the subset $J$ is an ideal of the ring $R$.

**(b)** Prove that the quotient ring $R/J$ is isomorphic to $\Q$.

Let $R$ be the ring of all $2\times 2$ matrices with integer coefficients:

\[R=\left\{\, \begin{bmatrix}

a & b\\

c& d

\end{bmatrix} \quad \middle| \quad a, b, c, d\in \Z \,\right\}.\]

Let $S$ be the subset of $R$ given by

\[S=\left\{\, \begin{bmatrix}

s & 0\\

0& s

\end{bmatrix} \quad \middle | \quad s\in \Z \,\right\}.\]

**(a)** True or False: $S$ is a subring of $R$.

**(b)** True or False: $S$ is an ideal of $R$.

Give 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 laterProve that the quadratic integer ring $\Z[\sqrt{5}]$ is not a Unique Factorization Domain (UFD).

Add to solve laterProve that the quadratic integer ring $\Z[\sqrt{-5}]$ is not a Unique Factorization Domain (UFD).

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 laterProve the following statements.

**(a)** If $a\neq 1$ is an idempotent element of $R$, then $a$ is a zero divisor.

**(b)** Suppose that $R$ is an integral domain. Determine all the idempotent elements of $R$.

Prove that the ring of integers

\[\Z[\sqrt{2}]=\{a+b\sqrt{2} \mid a, b \in \Z\}\]
of the field $\Q(\sqrt{2})$ is a Euclidean Domain.