## 1995 November

Instructions: Ph.D. students should attempt at least three problems, M.A. students should attempt at least two.

It goes without saying that all rings have an identity. Integral domains are commutative, other rings are not necessarily commutative unless so specified.

1. Recall that an element $r$ in an integral domain $R$ is called irreducible if whenever $r = ab$ for $a, b \in R$, then either $a$ or $b$ must have an inverse. (Actually, that's not quite the right definition; we should also assume $r$ is a nonzero nonunit element of $R$.)
a. Prove that an element $r$ is irreducible if and only if $(r)$ is maximal among principal ideals; i.e., $(r)$ is not properly contained in any other principal ideal.
b. Prove that a Noetherian integral domain $R$ is a unique factorization domain if and only if all principal ideals generated by irreducible elements are prime.

2. Let $R$ be a unique factorization domain. A polynomial $f\in R[X]$ is called primitive if $f \notin \mathfrak{a} R[X]$ for any proper ideal $\mathfrak{a}$ of $R$.
a. Prove that a polynomial $f$ is primitive if and only if $f \notin \mathfrak{p} R[X]$ for any prime ideal $\mathfrak{p}$ of $R$.
b. Prove that the product of two primitive polynomials is primitive. (This is essentially Gauss's Lemma.)
c. Show, in outline, how to use Gauss's Lemma to prove that the polynomial ring $R[X]$ is a unique factorization domain.

3. Prove that an integral domain $R$ has the property that every submodule of a free $R$-module is free if and only if $R$ is a principal ideal domain.

4. Let $I$ and $J$ be ideals in a commutative ring $R$ such that $I+J = R$.
a. Prove that $IJ = I\cap J$.
b. Prove that $R/IJ \cong R/I \oplus R/J$.
c. Prove that $I\oplus J \cong R \oplus IJ$.

5. a. Prove that a ring $R$ is left Noetherian if and only if all submodules of finitely generated left $R$-modules are finitely generated.
b. Prove that over any ring, a direct summand of a finitely generated module is finitely generated. (I.e., if $M \oplus N$ is finitely generated, then $M$ is finitely generated.)
c. Prove that over any ring $R$, a module is finitely generated and projective if and only if it is isomorphic to a direct summand of a free $R$-module with finite rank (i.e., of the form $R^n$ for some finite $n$).

6. Let $M$ be a module over a commutative ring $R$. Let $\mathfrak{p}$ be a prime ideal in $R$ and let $M_{\mathfrak{p}}$ be the localization of $M$ at $\mathfrak{p}$.
a. Prove that for $m\in M$, $0 \neq \frac{m}{1} \in M_{\mathfrak{p}}$ if and only if $\mathrm{ann}(m) \subseteq \mathfrak{p}$, where $\mathrm{ann}(m) = \{r \in R \mid rm = 0\}$.
b. Prove that an $R$-module $M$ is non-trivial if and only if $M_{\mathfrak{m}} \neq 0$ for some maximal ideal $\mathfrak{m}$ of $R$.