## 2001 November 26

1. Let $R$ be a ring and $M$ a (left) $R$-module. Suppose that $M = K\oplus L$. In this problem, maps are written on the right, opposite the scalars that operate on the left.
a. Let $\phi \in \operatorname{Hom}_R(L, K)$. Show that $M = K \oplus L(1 + \phi)$ where $L(1+\phi) = \{x + x\phi : x \in L\}$
b. Suppose that $M = K \oplus L'$. show that there is $\phi \in \operatorname{Hom}_R(L, K)$ such that $L' = L(1 +\phi)$.
c. Prove that $L(1 + \phi) = L(1 + \psi)$ for $\phi, \psi \in \operatorname{Hom}_R(L, K)$ if and only if $\phi = \psi$.
d. Note that the previous parts establish a bijective correspondence between complementary summands of $K$ in $M = K \oplus L$ and $\operatorname{Hom}_R(L, K)$. Let $M$ be a $\mathbb Z$-module (i.e., an abelian group), and suppose $M = \langle a \rangle \oplus \langle b \rangle \oplus \langle c \rangle$, where $\langle a \rangle$ is cyclic of prime order $p$, $\langle b \rangle$ is cyclic of order $p^2$, and $\langle c \rangle$ is cyclic of order $p^3$. Count the different direct complements of $\langle b \rangle \oplus \langle c \rangle$ in $M$.

2. Let $p$ be a prime in $\mathbb Z$, and let $R = {a/p^n : a, n \in \mathbb Z}$.
a. Prove that $R$ is a PID, hence a unique factorization domain.
b. What are the primes and what are the units in $R$?
c. In case $p=3$, prove that $9$ is a greatest common divisor of $5$ and $7$.

3. Let $R$ be a ring and suppose that $I, J$ are (two-sided) ideals of $R$. Recall that $IJ = \left\{\sum_i e_i f_i : e_i \in I, f_i \in J\right\}$.
a. Prove that $IJ$ is an ideal of $R$ and $IJ \subseteq I \cap J$.
b. Assume now that $R$ is a PID, $I= Ra$, and $J = Rb$. Show that $IJ = RAB$ and that $IJ = I\cap J$ if and only if $ab$ is a least common multiple of $a$ and $b$.

4. Explain why $\left(\frac{\mathbb Z}{\mathbb 5Z}\right) [x] / (x^2 - 2)$ is a field.