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

By assumption, $x^n=x$ is true for $n=1, 2$.

Suppose that $x^k=x$ for some $k \geq 2$ (induction hypothesis).
Then we have
\begin{align*}
x^{k+1}&=xx^k\\
&=xx &&\text{by induction hypothesis}\\
&=x^2=x &&\text{by assumption.}
\end{align*}

Thus, we conclude that $x^n=x$ for any positive integer $n$ by induction.

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

Let $x$ be a nilpotent element in $R$. That is, there is a positive integer $n$ such that $x^n=0$.

It follows from part (a) that $x=x^n=0$.
Thus every nilpotent element in $R$ is $0$.

Nilpotent Element a in a Ring and Unit Element $1-ab$
Let $R$ be a commutative ring with $1 \neq 0$.
An element $a\in R$ is called nilpotent if $a^n=0$ for some positive integer $n$.
Then prove that if $a$ is a nilpotent element of $R$, then $1-ab$ is a unit for all $b \in R$.
We give two proofs.
Proof 1.
Since $a$ […]

A Ring Has Infinitely Many Nilpotent Elements if $ab=1$ and $ba \neq 1$
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 […]

Is the Set of Nilpotent Element an Ideal?
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.
Proof.
We give a counterexample.
Let $R$ be the noncommutative ring of $2\times 2$ matrices with real […]

If the Localization is Noetherian for All Prime Ideals, Is the Ring Noetherian?
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?
Proof.
The answer is no. We give a counterexample.
Let […]

The Zero is the only Nilpotent Element of the Quotient Ring by its Nilradical
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 $\frakN(R/\frakN(R))=0$.
Proof.
Let $r\in R$ and if $x:=r+\frakN(R) \in R/\frakN(R)$ is a nilpotent element of […]

A Ring is Commutative if Whenever $ab=ca$, then $b=c$
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.
Proof.
Let $x, y$ be arbitrary elements in $R$. We want to show that $xy=yx$.
Consider the […]

Ring Homomorphisms and Radical Ideals
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 […]

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