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