We claim that there is at most one solution $x$ in the ring $R$.
Suppose that we have two solutions $r, s \in R$. That is, we have
\[r+r=1 \text{ and } s+s=1.\]

Then we have $r+r=s+s$. Putting $a=r-s\in R$, we have
\[a+a=0.\]
Now we compute
\begin{align*}
0&=1\cdot 0 =(r+r)(a+a)\\
&=ra+ra+ra+ra\\
&=(r+r)a+r(a+a)\\
&=1\cdot a+r\cdot 0\\
&=a.
\end{align*}

Therefore we obtain $a=0$ and thus $r=s$.
It follows that the equation $x+x=1$ has only one solution (at most).

Ring is a Filed if and only if the Zero Ideal is a Maximal Ideal
Let $R$ be a commutative ring.
Then prove that $R$ is a field if and only if $\{0\}$ is a maximal ideal of $R$.
Proof.
$(\implies)$: If $R$ is a field, then $\{0\}$ is a maximal ideal
Suppose that $R$ is a field and let $I$ be a non zero ideal:
\[ \{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 $a$ […]

Ring of Gaussian Integers and Determine its Unit Elements
Denote by $i$ the square root of $-1$.
Let
\[R=\Z[i]=\{a+ib \mid a, b \in \Z \}\]
be the ring of Gaussian integers.
We define the norm $N:\Z[i] \to \Z$ by sending $\alpha=a+ib$ to
\[N(\alpha)=\alpha \bar{\alpha}=a^2+b^2.\]
Here $\bar{\alpha}$ is the complex conjugate of […]

A ring is Local if and only if the set of Non-Units is an Ideal
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 […]

Finite Integral Domain is a Field
Show that any finite integral domain $R$ is a field.
Definition.
A commutative ring $R$ with $1\neq 0$ is called an integral domain if it has no zero divisors.
That is, if $ab=0$ for $a, b \in R$, then either $a=0$ or $b=0$.
Proof.
We give two proofs.
Proof […]

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 […]

Rings $2\Z$ and $3\Z$ are Not Isomorphic
Prove that the rings $2\Z$ and $3\Z$ are not isomorphic.
Definition of a ring homomorphism.
Let $R$ and $S$ be rings.
A homomorphism is a map $f:R\to S$ satisfying
$f(a+b)=f(a)+f(b)$ for all $a, b \in R$, and
$f(ab)=f(a)f(b)$ for all $a, b \in R$.
A […]

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 […]