# Tagged: ring

## 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 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 526

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.

## Problem 524

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 $R$.

## Problem 503

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.

## Problem 494

Prove that the rings $\Z[x]$ and $\Q[x]$ are not isomoprhic.

## Problem 489

Suppose that $\alpha$ is a rational root of a monic polynomial $f(x)$ in $\Z[x]$.
Prove that $\alpha$ is an integer.

## Problem 417

Let $R$ be a ring with $1$ and let $M$ be an $R$-module. Let $I$ be an ideal of $R$.
Let $M’$ be the subset of elements $a$ of $M$ that are annihilated by some power $I^k$ of the ideal $I$, where the power $k$ may depend on $a$.
Prove that $M’$ is a submodule of $M$.

## Problem 416

Let $R$ be a ring with $1$. Let $M$ be an $R$-module. Consider an ascending chain
$N_1 \subset N_2 \subset \cdots$ of submodules of $M$.
Prove that the union
$\cup_{i=1}^{\infty} N_i$ is a submodule of $M$.

## Problem 415

(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.

## Problem 414

Let $R$ be a ring with $1$. Let
$0\to M\xrightarrow{f} M’ \xrightarrow{g} M^{\prime\prime} \to 0 \tag{*}$ be an exact sequence of left $R$-modules.

Prove that if $M$ and $M^{\prime\prime}$ are finitely generated, then $M’$ is also finitely generated.

## Problem 413

Suppose that $f:R\to R’$ is a surjective ring homomorphism.
Prove that if $R$ is a Noetherian ring, then so is $R’$.

## Problem 412

Let $f: R\to R’$ be a ring homomorphism. Let $P$ be a prime ideal of the ring $R’$.

Prove that the preimage $f^{-1}(P)$ is a prime ideal of $R$.

## Problem 411

Let $f:R\to R’$ be a ring homomorphism. Let $I’$ be an ideal of $R’$ and let $I=f^{-1}(I)$ be the preimage of $I$ by $f$. Prove that $I$ is an ideal of the ring $R$.

## Problem 410

Let $R$ be a ring with $1$ and let $M$ be a left $R$-module.
Let $S$ be a subset of $M$. The annihilator of $S$ in $R$ is the subset of the ring $R$ defined to be
$\Ann_R(S)=\{ r\in R\mid rx=0 \text{ for all } x\in S\}.$ (If $rx=0, r\in R, x\in S$, then we say $r$ annihilates $x$.)

Suppose that $N$ is a submodule of $M$. Then prove that the annihilator
$\Ann_R(N)=\{ r\in R\mid rn=0 \text{ for all } n\in N\}$ of $M$ in $R$ is a $2$-sided ideal of $R$.

## Problem 409

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.

## Problem 408

Let $R$ be a ring with $1$ and $M$ be a left $R$-module.

(a) Prove that $0_Rm=0_M$ for all $m \in M$.

Here $0_R$ is the zero element in the ring $R$ and $0_M$ is the zero element in the module $M$, that is, the identity element of the additive group $M$.
To simplify the notations, we ignore the subscripts and simply write
$0m=0.$ You must be able to and must judge which zero elements are used from the context.

(b) Prove that $r0=0$ for all $s\in R$. Here both zeros are $0_M$.

(c) Prove that $(-1)m=-m$ for all $m \in M$.

(d) Assume that $rm=0$ for some $r\in R$ and some nonzero element $m\in M$. Prove that $r$ does not have a left inverse.

## Problem 345

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 $R/I$.