Homework Rings and Modules, B.Math


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

    Homework 1

  1. Show that $(\mathbb Z/n\mathbb Z)$ contains a nonzero nilpotent element iff $p^2$ divides $n$ for some prime $p$ (i.e. $n$ is not squarefree).

  2. Let $u$ be a unit and $a$ be a nilpotent element of a ring $R$. Show that $u+a$ is a unit in $R$.

  3. Let $G$ be a nontrivial finite group. Let $RG$ be the group ring of $G$ over $\mathbb C$. Show that the center of $RG$ (i.e. {$x\in RG: xy=yx$ for all $y\in RG$}) is strictly bigger that $\mathbb{C} e$ where $e$ is the identity element of $G$.

    Determine whether the following pair of rings are isomorphic. Justify your answer.

  4. $(\mathbb{Z}/p\mathbb{Z})[x]$ and $\mathbb Z$.
  5. $\mathbb{Q}[x]$ and $\mathbb{Z}[x]$.
  6. $\mathbb{C}$ and $R_{\mathbb R}G$ where $G$ is a group of order 2 and $R_{\mathbb R}G$ is the group ring over $\mathbb R$.

    Homework 2

  7. Let $n=p_1^{r_1}p_2^{r_2}\ldots p_k^{r_k}$ be an integer with $p_i$ prime numbers. Find all prime ideals of $\bfrac{\mathbb{Z}}{n\mathbb{Z}}[x]$ containing monic polynomials of degree one.

  8. An element $a$ in a ring $R$ is called nilpotent if $a^n=0$ for some $n\ge 1$. Show that if $u$ is a unit in $R$ and $a$ is nilpotent then $u+a$ is a unit.

  9. Dummit and Foote, Section 7.1 Exercise 26.

  10. Dummit and Foote, Section 7.1 Exercise 27.

  11. Dummit and Foote, Section 7.2 Exercise 5.

  12. Dummit and Foote, Section 7.3 Exercise 33.

  13. Dummit and Foote, Section 7.4 Exercise 9.

  14. Dummit and Foote, Section 7.4 Exercise 10.