An element $x\in R$ is called a zero divisor if there exists a nonzero element $y\in R$ such that $xy=0$ or $yx=0$.
So if $x$ is not a zero dividor, then $xy=0$ implies that $y=0$. Similarly, $yx=0$ implies that $y=0$.
Proof.
(a) Prove that if $a$ is not a zero divisor, then $ba=1$.
Suppose that $a$ is not a zero divisor. We compute
\begin{align*}
a(ba-1)&=aba-a && \text{by distributivity}\\
&=1\cdot a -a &&\text{by $ab=1$}\\
&=a-a=0.
\end{align*}
Since $a$ is not a zero divisor, this yields that $ba-1=0$, and hence $ba=1$.
(b) Prove that if $b$ is not a zero divisor, then $ba=1$.
Suppose that $b$ is not a zero divisor. We calculate
\begin{align*}
(ba-1)b&=bab-b && \text{by distributivity}\\
&=b\cdot 1 -b &&\text{by $ab=1$}\\
&=b-b=0.
\end{align*}
As $b$ is not a zero divisor, the equality $(ba-1)b=0$ implies that $ba-1=0$.
Hence we have $ba=1$.
No Nonzero Zero Divisor in a Field / Direct Product of Rings is Not a Field
(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.
Proof.
(a) Show that $F$ does not have a nonzero zero divisor.
[…]
If Every Proper Ideal of a Commutative Ring is a Prime Ideal, then It is a Field.
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.
Proof.
As the zero ideal $(0)$ of $R$ is a proper ideal, it is a prime ideal by assumption.
Hence $R=R/\{0\}$ is an integral […]
Is the Quotient Ring of an Integral Domain still an Integral Domain?
Let $R$ be an integral domain and let $I$ be an ideal of $R$.
Is the quotient ring $R/I$ an integral domain?
Definition (Integral Domain).
Let $R$ be a commutative ring.
An element $a$ in $R$ is called a zero divisor if there exists $b\neq 0$ in $R$ such that […]
If a Prime Ideal Contains No Nonzero Zero Divisors, then the Ring is an Integral Domain
Let $R$ be a commutative ring. Suppose that $P$ is a prime ideal of $R$ containing no nonzero zero divisor. Then show that the ring $R$ is an integral domain.
Definitions: zero divisor, integral domain
An element $a$ of a commutative ring $R$ is called a zero divisor […]
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 […]
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\} […]
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 […]
The Preimage of Prime ideals are Prime Ideals
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 $R$.
Proof.
The preimage of an ideal by a ring homomorphism is an ideal.
(See the post "The inverse image of an ideal by […]