## Is the Set of Nilpotent Element an Ideal?

## 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.

Add to solve laterIs 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.

Add to solve laterLet $R$ 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.

Add to solve laterLet $R$ 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 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)** 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.

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 $R$ 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 $R$.

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.

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

Prove that $\alpha$ is an integer.

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 $M$.

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 $M$.

**(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.

Read solution

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.

Add to solve later Suppose that $f:R\to R’$ is a surjective ring homomorphism.

Prove that if $R$ is a Noetherian ring, then so is $R’$.

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 $R$.

Add to solve laterLet $f:R\to R’$ 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 $R$.

Add to solve later 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 $R$.