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 $\frakN(R/\frakN(R))=0$.
Add to solve later Let $R$ be a principal ideal domain. Let $a\in R$ be a nonzero, non-unit element. Show that the following are equivalent.
(1) The ideal $(a)$ generated by $a$ is maximal.
(2) The ideal $(a)$ is prime.
(3) The element $a$ is irreducible.
Add to solve later Let $R$ be a finite commutative ring with identity $1$. Prove that every prime ideal of $R$ is a maximal ideal of $R$.
Add to solve later 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 surjective and $\ker(f)\subset I$. Then prove that $f(\sqrt{I}\,) =\sqrt{f(I)}$
Add to solve later Let $I=(x, 2)$ and $J=(x, 3)$ be ideal in the ring $\Z[x]$.
(a) Prove that $IJ=(x, 6)$.
(b) Prove that the element $x\in IJ$ cannot be written as $x=f(x)g(x)$, where $f(x)\in I$ and $g(x)\in J$.
Add to solve later 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 later Let $R$ be a commutative ring with $1$ such that every element $x$ in $R$ is idempotent, that is, $x^2=x$. (Such a ring is called a Boolean ring.)
(a) Prove that $x^n=x$ for any positive integer $n$.
(b) Prove that $R$ does not have a nonzero nilpotent element.
Add to solve later 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?
Add to solve later 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.
Add to solve later 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 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 Let $\Z[x]$ be the ring of polynomials with integer coefficients.
Prove that
\[I=\{f(x)\in \Z[x] \mid f(-2)=0\}\]
is a prime ideal of $\Z[x]$. Is $I$ a maximal ideal of $\Z[x]$?
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.
Add to solve later 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 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) Prove that every prime ideal of a Principal Ideal Domain (PID) is a maximal ideal.
(b) Prove that a quotient ring of a PID by a prime ideal is a PID.
Add to solve later Let $I$ be a nonzero ideal of the ring of Gaussian integers $\Z[i]$.
Prove that the quotient ring $\Z[i]/I$ is finite.
Add to solve later Let $R$ and $S$ be rings. Suppose that $f: R \to S$ is a surjective ring homomorphism.
Prove that every image of an ideal of $R$ under $f$ is an ideal of $S$.
Namely, prove that if $I$ is an ideal of $R$, then $J=f(I)$ 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.
Add to solve later Let $R$ be a commutative ring with identity $1\neq 0$. Suppose that for each element $a\in R$, there exists an integer $n > 1$ depending on $a$.
Then prove that every prime ideal is a maximal ideal.
Add to solve later 
