Algebra II - May 2023

Instructions

You have 120 minutes to complete this exam.

Problem 1.

(10 points) Let $A$ be an integral domain and let $M$ be a finitely generated torsion module over $A$ . Prove that there is a nonzero element $a \in A$ such that $a m = 0$ for all $m \in M$ .

Problem 2.

(10 points) Give an example of three modules $A$ , $B$ and $C$ over a principal ideal domain (PID) such that the sequence

0 \to A \to B \to C \to 0

is exact, but $B$ is not isomorphic to $A \oplus C$ .

Problem 3.

(10 points) Prove that the quotient group $R / Z$ is an injective abelian group.

Problem 4.

(10 points) Note that the set of $2 \times 2$ upper triangular matrices

A := {(\begin{matrix} a & b \\ 0 & c \end{matrix}) ∣ a, b, c \in C}

forms a ring under the usual matrix addition and multiplication. Consider the subsets

M_{1} := {(\begin{matrix} a & 0 \\ 0 & 0 \end{matrix}) ∣ a \in C},

M_{2} := {(\begin{matrix} 0 & b \\ 0 & c \end{matrix}) ∣ b, c \in C} .

Prove that $M_{1}$ and $M_{2}$ are both left ideals of $A$ .
Prove that $M_{1}$ and $M_{2}$ are both projective left modules over $A$ .

Problem 5.

(10 points) Compute the group

{Ext}_{Z}^{1} (Z / 12, Z / 18) .

Problem 6.

(10 points) Recall the principal ideal domain of Gaussian integers

Z [\sqrt{- 1}] := {a + b \sqrt{- 1} ∣ a, b \in Z} .

Let $J \in G L_{n} (Z)$ be an integer matrix such that

J^{2} = - I_{n},

where $I_{n}$ is the identical matrix. We view $Z^{n}$ as a module over $Z [\sqrt{- 1}]$ such that

\forall a + b \sqrt{- 1} \in Z [\sqrt{- 1}], \forall \vec{v} \in Z^{n}, (a + b \sqrt{- 1}) \cdot \vec{v} := a \vec{v} + b J \vec{v} .

Prove that $Z^{n}$ is a free module over $Z [\sqrt{- 1}]$ .
Determine the rank of $Z^{n}$ as a module over $Z [\sqrt{- 1}]$ .