Let $R$ be a commutative ring with unity. Let $J \subset I$ be ideals of $R$. Let $\bar I$ denote the ideal $I / J$ of $\bar R=R/J$. Show that $\bar R/\bar I$ is isomorphic to $R/I$.
Let $R$ be a ring. Show that nilrad($R$)$=\cap\{P: P \text{ prime ideal of }R\}$.
An element $x$ in a ring $R$ is called idempotent if $x^2=x$. Show that the only idempotents in $\mathbb{R}[X,Y]/(XY)$ are 0 and 1.
Show that $R$ is a local ring iff for any $x\in R$, $x$ or $1+x$ is a unit.
Let $p$ be a prime number. Show that the subring $\{r\in \mathbb{Q}: r=\frac{m}{n}, m,n \in \mathbb{Z} \text{ and } p \nmid n \}$ of $\mathbb{Q}$ is a local ring.
Properties of V,I on Spec. Let $R$ be a ring.
(a) Show that $I\subset J$ ideals of $R$ implies $V(J)\subset V(I)$ and $Y\subset X \subset Spec(R)$ implies $\mathscr{I}(X)\subset \mathscr{I}(Y)$.
(b) Show that $V(\cup I_{\alpha})=\cap V(I_{\alpha})$ for ideals $I_{\alpha}$.
(c) Show that $V(IJ)=V(I\cap J)=V(I)\cup V(J)$ for ideals $I, J$.
(d) $\mathscr{I}(V(I))=\sqrt{I}$.
Show that $I$ is a prime ideal of $R$ then $V(I)$ is irreducible.
Show that $Spec(R)$ is connected iff only idempotent in $R$ are 0 and 1.
(Complete the construction of localization $S^{-1}R$) Verify the sum in $S^{-1}R$ is well defined and check associativity.
Show that $S^{-1}R=0$ iff $0\in S$. Show that the natural map $R\to S^{-1}R$ is injective iff $S$ does not contain a zero divisor. The map is an isomorphism iff every element of $S$ is a unit.
Show that the contraction and extension of ideals between $R$ and $S^{-1}R$induces a bijection between prime ideals of $R$ disjoint from $S$ and prime ideals of $S^{-1}R$.
Let $R=\mathbb{C}[X,Y]$. Show that the ideal $I=(X,Y)$ is not a free $R$-module.
Let $M$ be an $R$-module and $\mu:R\to End(M)$ the resulting structure map. Show that $End_R(M)$ is the centralizer of $Image(\mu)$.
Let $M$ be an $R$-module and $\phi:M\to M$ an $R$-linear map. Show that the action of $R[x]$ on $M$ defined in the class makes $M$ an $R[x]$-module.
Let $R$ be a ring and $S$ a multiplicative subset. Show that $S^{-1}R$ is naturally an $R$-module. Show that it is not finitely generated if $S$ consists of nonzero divisors and at least one non-unit.
Let $R$ be a ring, $M$ a finitely generated $R$-module and $S$ amultiplicative subset of $R$. Show that $S^{-1}M$ is a finitely genrated $S^{-1}R$-module and if $M$ is a free $R$-module then $S^{-1}M$ is a free $S^{-1}R$-module. Is $S^{-1}M$ a finitely generated $R$-module?
Let $R=k[x,y]$ and $M=k[x,y]/(x)$. Compute $S^{-1}M$ for
(a) $S=R\setminus (x)$.
(b) $S=\{1, y, y^2, \ldots \}$.
Let $R$ be a ring, $S$ a multiplicative subset of $R$ and $f:A\to B$ a $R$-linear map. Show that the map $S^{-1}f:S^{-1}A\to S^{-1}B$ sending $a/s$ to $f(a)/s$ is a well defined and $S^{-1}R$-linear map.
Show that every short exact sequence of $k$-module splits. Give an example of a short exact sequence of $R$-modules for some ring $R$ which does not split.
Let $0\to A\to B \to C\to 0$ be a short exact sequence of $R$-modules. Show that if $A$ and $C$ are finitely generated then so is $B$. But the converse fails ($B$ is finitely generated does not imply $A$ is finitely generated).
Let $\mathscr{C}$ be an abelian category and let $f\in Mor(A,B)$ for some objects $A$ and $B$ of $\mathscr{C}$. Write down the universal property an object $I$ together with $i\in Mor(I,B)$ should satisfy for it to be image of $f$. Show that kernel(cokernel($F$)) satisfy this universal property.
Let $0\to A\to B \to C\to 0$ be a short excat sequence of $R$-modules and $M$ be an $R$-module. Show that $0\to Hom(C,M)\to Hom(B,M)\to Hom(A,M)$ is an exact sequence. Show that $Hom(B,M)\to Hom(A,M)$ need not be surjective.
Show that for a module $M$, $Tor(M)$ the subset of torsion elements of $M$, need not be a submodule of $M$.
Let $R$ be a ring and $M$, $N$ and $K$ be $R$-modules. Show that $Hom(M,N\oplus K)$ is isomorphic to $Hom(M,N)\oplus Hom(M,K)$ as $R$-modules.
Show that for a field $k$, $k[X]\otimes_k k[Y]\cong k[X,Y]$.
Let $R$ be a ring and $M$ an $R$-module. Show that $-\otimes_R M$ need not be an exact functor.
Let $(M_i)_{i\in \Omega}$ be $R$-modules and $M=\oplus_{i\in \Omega}M_i$. Show that $M$ is flat iff $M_i$ is flat for all $i\in \Omega$.
Let $R$ be a ring and $S$ be a multiplicative subset. Let $M$ be a finitely generated $R$-module. Show that $S^{-1}Ann(M)=Ann(S^{-1}M)$ as ideals of $S^{-1}R$.
Let $M$ be an $R$-module. Show that the following are equivalent.
(i) $M=0$.
(ii) $M_P=0$ for all prime ideals $P$ in $R$.
(iii) $M_m=0$ for all maximal ideals $m$ in $R$.
Let $B$ be an integral domain and $A$ a subring of $B$. Let $K$ be the field of fractions of $A$. Show that if $\alpha \in B$ is integral over $A$ then every $K$-Galois conjugate of $\alpha$ (which lies in an algebraic closure of the quotient field of $B$) is integral over $A$.
Let $A\subset B$ be an integral extension. Show that $A[X]\subset B[X]$ is also an integral extension.
Let $A\subset B$ be rings and $S\subset A$ be a multiplicative set. Let $C$ be the integral closure of $A$ in $B$. Show that $S^{-1}C$ is the integral closure of $S^{-1}A$ in $S^{-1}B$.
Show that the ring $k[X,Y]/(Y^2-X^2-X^3)$ is not a normal domain.
Show that $\mathbb{Z}[i]$ is the integral closure of $\mathbb{Z}$ in $\mathbb{Q}[i]$.
Let $k$ be a field and $R$ be a finitely generated $k$-algebra. Show that $Jac(R)=nil(R)$.
Let $A\subset B$ be a finitely generated $k$-algebras. Show that for any maximal ideal $n$ of $B$, $B/n$ is a finite extension of $A/(A\cap n)$.
Let $A\subset B$ be integral extension. Show that dimension of $A$ and $B$ are same.
Find an example of a ring extension $A \subset B$, a prime ideal $P$ in $A$ and prime ideals $Q_1\subset Q_2$ in $B$ such that $Q_1\cap A=Q_2\cap A=P$.
Find an example of a ring extension $A \subset B$ such that the induced map from Spec$(B)$ to Spec$(A)$ is surjective (i.e. lying over holds) but going up property for the extension fails.
Let $A\subset B$ be integral extension and $P$ a prime ideal of $A$. Show that $PB\cap A=P$.
Let $A=k[X,Y,Z]/(Y^2-X^2-X^3)$ and $x,y,Z$ be the images of $X,Y,Z$ in $R$. Let $B=A[y/x]$. Show that $B=k[y/x,Z]$ is isomorphic to the polynomial ring in two variables and hence $Q_1=(y/x+1,Z)$ is a prime ideal of $B$. Show that $Q_1\cap A=(x,y,Z)$ (=$P_1$ say). Show that for any nonzero $\alpha\in k$, $(y/x-1+\alpha Z)\cap A=(y-x+\alpha xZ)A$ (=$P_0$ say). Show that $P_0B=(y/x-1+\alpha Z)$ and hence is not contained in $Q_1$. Deduce that going down does not hold for the integral extension $B/A$.
Let $A\subset B$ be an integral extension. Let $P\in spec(A)$ and $Q\in spec(B)$. Show that $Q$ is a minimal prime of $PB$ iff $Q\cap A=P$.
Let $A$ be a ring and $S$ a multiplicative subset. Show that going down theorem holds for the following ring homomorphisms.
(i) $A\subset A[X]$ where $A[X]$ is the polynomial ring over $A$.
(ii) $A \to S^{-1}A$ where $S$ is a multiplicative subset of $A$.
(More generally, going down holds for the extension $A\subset B$ if $B$ is a flat $A$-module.)
(Power of prime ideals need not be primary): Let $R=k[x,y,z]/(z^2-xy)$. Show that $P=(\bar x, \bar z)$ is a prime ideal of $R$. Show that $P^2$ is not primary. Also show $\sqrt{P^2}=P$. (Hint: First show $\bar x\bar y \in P^2$)
Let $R$ be a ring and $I$ an ideal in $R$. Let $R[X]$ be a polynomial ring over $R$ and $I[X]$ the ideal generated by $I$ in $R[X]$. Show that if $I$ is primary then $I[X]$ is primary. (Hint: Show that zero divisors in $R[X]/I[X]\cong (R/I)[X]$ and zerodivisors in $(R/I)[X]$ are nilpotents.)
Let $k$ be a field and $k[X_1,\ldots, X_n]$ be a polynomial ring. Show that every power of prime ideals $(X_1,\ldots, X_i)$ is a primary ideal for $1\le i \le n$. (Hint: Use above)
Let $R=k[X,Y,Z]$ and $I=(X^2,XY,YZ,ZX)$. Determine the minimal prime ideals of $I$. Show that $(X,Y,Z)$ is radical of a primary ideal which appear in a minimal primary decomposition of $I$.
Let $R$ be a ring and $S$ a multiplicative subset of $R$. Let $Q$ be a primary ideal of $R$ disjoint with $S$ then show that $S^{-1}Q$ is a primary ideal of $S^{-1}R$.
Let $R=k[X,Y,Z]$ and $I=(X^2,XY,YZ,ZX)$. Determine the minimal prime ideals of $I$. Show that $(X,Y,Z)$ is an embedded prime of $I$.
Let $R$ be a ring and $M$ a simple $R$-module then show that $M$ is isomorphic to $R/m$ for some maximal ideal $m$ of $R$.
Let $A$ be a noetherian ring. Prove that the following are equivalent.
(i) $A$ is Artinian.
(ii) Spec($A$) is finite and discrete.
(iii) Spec($A$) is discrete.
Let $k$ be a field and $A$ be a finitely generated $k$-algebra. Show that $A$ is artinian iff $dim_k(A)< \infty$.
Let $v:K^*\to G$ be a valuation where $K$ is a field and $G$ an ordered abelian group. Show that the valuation ring $R_v$ is noetherian iff $Im(v)$ is isomorphic to $\mathbb{Z}$.
Let $A$ be a Dedekind domain and $I$ a nozero ideal in $A$. Show that every ideal in $A/I$ is principal. Deduce that every ideal in $A4 can be generated by two elements.
Let $M$ be a finitely generated module over a dedekind domain. Show that $M$ is flat iff $M$ is torsion free.
Let $(R,m)$ be a noetherian local ring such that $m$ is a principal ideal and $R$ is not artinian. Show that $R$ is a discrete valuation ring.
Let $\hat{\mathbb{Z}}_p$ be the inverse limit of $\mathbb{Z}/p^n\mathbb{Z}$ as $n$ varies. Show that every element of $\hat{\mathbb{Z}}_p$ can be uniquely written as a power series:
$ a_0 + a_1p + a_2p^2 + \ldots $ where $a_i\in \{0, 1, \ldots, p-1\}$ and addition happens with carry over, i.e. $3p+(p-2)p = 1p + 1p^2$.
Let $A$ be noetherian ring $I$ an ideal and $\hat A$ the $I$-adic completion of $A$. Show that if $x\in A$ is a non-zero divisor then its image in $\hat A$ is a no0n-zero divisor. Consider the local ring $A=\mathbb{C}[x,y]_{(x,y)}/(x^2-y^2-y^3)$ and let $m$ denote its maximal ideal. Show that the $m$-adic completion of $A$ is not an integral domain.
Let $(R,m)$ be a DVR. Show that its $m$-adic completion $\hat R$ is also a DVR.