Ring Theory Exam

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2004 November

Notation: $\mathbb Z$ denotes the ring of integers and $\mathbb Q$ denotes the field of rational numbers.

  1. Let $R$ be a commutative ring.
    a. Show that if $R$ is an integral domain, then the only units in the polynomial ring $R[x]$ are the units of $R$.
    b. Give counterexamples when $R$ is not an integral domain.
    c. Show that $R[x]$ is a principal ideal domain if and only if $R$ is a field.

  2. Let $f\in \mathbb Z[x]$ be a monic polynomial of degree $n$ with distinct roots $\alpha_1, \dots, \alpha_r$, $r \leq n$. Show that $\alpha_1 + \cdots + \alpha_r \in \mathbb Z$.

  3. List all the ideals of the quotient ring $\mathbb R[x]/I$, where $I$ is the ideal generated by $(x-5)^2(x^2 + 1)$. Identify which of the ideals are prime and which are maximal. Does your answer change if the field $\mathbb R$ is replaced by $\mathbb C$? (Explain.)

  4. Let $M$ be a left module over a ring $R$.
    a. Suppose that $M$ is finitely generated and that $R$ is commutative and Noetherian. Sketch a proof that $M$ is Noetherian.
    b. Suppose that $M$ is Noetherian and that $f: M \rightarrow M$ is a surjective homomorphism. Show that $f$ is an isomorphism.

  5. Let $R$ be a ring with $1$ and let $M$ be an $R$-module. Show that the following are equivalent:
    i. There exists a module $N$ such that $M\oplus N$ is free.
    ii. Given any surjection $\varphi: B\rightarrow M$, there exists an $R$-module homomorphism $\psi: M \rightarrow B$ such that $\varphi \circ \psi$ is the identity on $M$.
    iii. Given a homomorphism $\varphi: M\rightarrow B$ and a surjection $\pi: A \rightarrow B$, there exists a homomorphism $\psi : M \rightarrow A$ such that $\pi \circ \psi = \varphi$.

  6. Let $R = \mathbb Z[i]$ be the ring of Gaussian integers.
    a. Show that any nontrivial ideal must contain some positive integer.
    b. Find all the units in $R$.
    c. If $a + bi$ is not a unit, show that $a^2 + b^2 > 1$.