0;10;1c Homework Commutative Algebra MMath and BMath

Homework for Commutative Algebra I, M.Math and B.Math


Unless otherwise stated a ring would mean a commutative ring with identity.

    Week 1

  1. 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 $R/I$.

  2. An element $x$ in a ring $R$ is called idempotent if $x^2=x$. Show that if a ring $R$ has an idempotent different from 0 and 1 then $R$ is not an integral domain. Show that the only idempotents in $\mathbb{R}[X,Y]/(XY)$ are 0 and 1.

  3. 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.

  4. 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 $Image(\mu)$.

  5. Show that $R$ is a local ring iff for any $x\in R$, $x$ or $1+x$ is a unit.

  6. 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.

    Week 2

  7. Let $R=k[x,y]$ and $M=k[x,y]/(x)$. Compute $S^{-1}M$ for
    (a) $S=R\setminus (x)$.
    (b) $S=\{1, y, y^2, \ldots \}$.

  8. Show that $Spec(R)$ is connected iff only idempotent in $R$ are 0 and 1.

  9. Let $R$ be a ring. For a subset $X$ of $Spec(R)$, define $I(X)=\cap_{P\in X} P$.
    (a) Show that $Y\subset X \subset Spec(R)$ implies $\mathscr{I}(X)\subset \mathscr{I}(Y)$.
    (b) $\mathscr{I}(V(I))=\sqrt{I}$.

  10. Show that $I$ is a prime ideal of $R$ then $V(I)$ is irreducible.

  11. 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.

  12. 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.

    Week 3 and 4

  13. Let $R=\mathbb{C}[X,Y]$. Show that the ideal $I=(X,Y)$ is not a free $R$-module.

  14. 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?

  15. 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).

  16. 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.

  17. Prove the Five Lemma. Let the following be a commutative diagram of $R$-modules with exact rows: $$ \require{AMScd} \begin{CD} M_1 @>{f_1}>> M_2 @>{f_2}>> M_3 @>{f_3}>> M_4 @>{f_4}>> M_5 \\ @V{h_1}VV @V{h_2}VV @V{h_3}VV @V{h_4}VV @V{h_5}VV \\ N_1 @>>{g_1}> N_2 @>>{g_2}> N_3 @>>{g_3}> N_4 @>>{g_4}> N_5 \end{CD} $$ 1. If $h_1$ is surjective, and $h_2, h_4$ are injective, then $h_3$ is injective.
    2. If $h_5$ is injective, and $h_2, h_4$ are surjective, then $h_3$ is surjective.
    3. If $h_1, h_2, h_4, h_5$ are isomorphisms, then $h_3$ is an isomorphism.

  18. Let $k$ be a field and $R=k[[x,y]]/(x^2+xy)$. Let $M=R/(x)$. Show that $M$ is not a projective $R$-module. Also show that $M/xM$ is a free $R/(x)$-module.

  19. In the proof of construction of tensor product, show that the map from $\tilde \psi:T \to Q$ satisfying $\tilde\psi \circ \phi=\psi$ is unique.

  20. Let $R$ be a ring and $A, B, C$ be $R$-modules. Show that $(A\otimes B)\otimes C \cong A\otimes (B\otimes C)$.

    Week 5

  21. Let $R$, $A$ and $B$ be rings. Let $i:R\to A$ and $j:R\to B$ be ring homomorphisms. Show that there exist an $R$-bilinear map $\mu:(A\otimes_R B) \times (A\otimes_R B) \to A\otimes_R B$ which sends $(a\otimes b, a'\otimes b')$ to $aa'\otimes bb'$. Show that this makes $A\otimes_R B$ into a ring.

  22. Show that for a field $k$, $k[X]\otimes_k k[Y]\cong k[X,Y]$.

  23. Let $A$ be a ring, $B$ be an $A$-algebra and $M$ a finitely generated $B$-module. Show that $M$ is a finitely generated $A$-module if $B$ is a finitely generated $A$-module.

  24. Let $0\to A\to B\to C\to 0$ be an exact sequence of $R$-modules. Show that if $A$ and $C$ are noetherian $R$-module then so is $B$.

  25. Show that if $M$ is a flat $R$-module and $S$ is a multiplicative subset of $R$ then $S^{-1}M$ is a flat $S^{-1}R$-module.

  26. 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 $i\in \Omega$.

    Week 6

  27. Let $R$ be a ring, $M$ and $N$ be $R$-modules and $A$ be an $R$-algebra. Show that $A\otimes_R(M\otimes_R N)\cong (A\otimes_R M)\otimes_A (A\otimes_R N)$ as $A$-modules.

  28. Let $M$ be flat $R$-module where $R$ is an integral domain. Show that $M$ is torsion free.

  29. Let $R=\mathbb C[x,y]$. Show that the ideal $I=(x,y)$ is not flat. (Hint: Show that the natural map $I\otimes I\to I$ is not injective by showing that $x\otimes y-y\otimes x$ is a non zero element in $I\otimes I$ and is in the kernel of the map.)