# Category: Ring theory

## Problem 725

Prove that if $R$ is a commutative ring and $\frakN(R)$ is its nilradical, then the zero is the only nilpotent element of $R/\frakN(R)$. That is, show that $\frakN(R/\frakN(R))=0$.

## Problem 724

Let $R$ be a principal ideal domain. Let $a\in R$ be a nonzero, non-unit element. Show that the following are equivalent.

(1) The ideal $(a)$ generated by $a$ is maximal.
(2) The ideal $(a)$ is prime.
(3) The element $a$ is irreducible.

## Problem 723

Let $R$ be a finite commutative ring with identity $1$. Prove that every prime ideal of $R$ is a maximal ideal of $R$.

## Problem 624

Let $R$ and $R’$ be commutative rings and let $f:R\to R’$ be a ring homomorphism.
Let $I$ and $I’$ be ideals of $R$ and $R’$, respectively.

(a) Prove that $f(\sqrt{I}\,) \subset \sqrt{f(I)}$.

(b) Prove that $\sqrt{f^{-1}(I’)}=f^{-1}(\sqrt{I’})$

(c) Suppose that $f$ is surjective and $\ker(f)\subset I$. Then prove that $f(\sqrt{I}\,) =\sqrt{f(I)}$

## Problem 623

Let $I=(x, 2)$ and $J=(x, 3)$ be ideal in the ring $\Z[x]$.

(a) Prove that $IJ=(x, 6)$.

(b) Prove that the element $x\in IJ$ cannot be written as $x=f(x)g(x)$, where $f(x)\in I$ and $g(x)\in J$.

## Problem 620

Is it true that a set of nilpotent elements in a ring $R$ is an ideal of $R$?

If so, prove it. Otherwise give a counterexample.

## Problem 618

Let $R$ be a commutative ring with $1$ such that every element $x$ in $R$ is idempotent, that is, $x^2=x$. (Such a ring is called a Boolean ring.)

(a) Prove that $x^n=x$ for any positive integer $n$.

(b) Prove that $R$ does not have a nonzero nilpotent element.

## Problem 617

Let $R$ be a commutative ring with $1$.
Suppose that the localization $R_{\mathfrak{p}}$ is a Noetherian ring for every prime ideal $\mathfrak{p}$ of $R$.
Is it true that $A$ is also a Noetherian ring?

## Problem 615

Let $R$ be a ring and assume that whenever $ab=ca$ for some elements $a, b, c\in R$, we have $b=c$.

Then prove that $R$ is a commutative ring.

## 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.

## 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?

## Problem 573

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 $\Z[x]$?

## Problem 543

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.

## Problem 542

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 $ba=1$.

## Problem 536

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 $S$.

## Problem 535

(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.

## Problem 534

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

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

## Problem 532

Let $R$ 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 $S$.

## Problem 531

(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.

## Problem 530

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.