If Every Proper Ideal of a Commutative Ring is a Prime Ideal, then It is a Field.
Problem 598
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.
Add to solve later Tagged: zero divisor
Add to solve later Is the Quotient Ring of an Integral Domain still an Integral Domain?
Problem 589
Let $R$ be an integral domain and let $I$ be an ideal of $R$.
Is the quotient ring $R/I$ an integral domain?
Add to solve later If $ab=1$ in a Ring, then $ba=1$ when $a$ or $b$ is Not a Zero Divisor
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 $ba=1$.
Add to solve later No Nonzero Zero Divisor in a Field / Direct Product of Rings is Not a Field
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.
Add to solve later Torsion Submodule, Integral Domain, and Zero Divisors
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.
Add to solve later If a Prime Ideal Contains No Nonzero Zero Divisors, then the Ring is an Integral Domain
Problem 220
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.
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