Exercises:Rings and Modules - 2016 - 1

Problems

Problem 1

Part A

Let $R$ be a u-ring. Fix an $a\in R$ and define a homomorphism:

• $\varphi_a:R[T]\rightarrow R$ by $\varphi_a:P(T)\mapsto P(a)$ - evaluation at $a$.

By restriction of scalars every $\varphi_a$ gives the target $R$ the structure of an $R[T]$-module, which we denote $R_a$.

Show that for $a,b\in R$ that:

• there is an $R[T]$-module isomorphism between $R_a$ and $R_b$

if and only if

• $a=b$

Part B

Let $M$ be an $R$-module. Show that there is a surjection from a free $R$-module onto $M$.

Part C

Show that the $\mathbb{Z}$-module, $\mathbb{Q}$, is not free.

Problem 2

Part A

Compute the homology groups at $\mathbb{Q}^5$ and $\mathbb{Z}^5$ of the the following complexes:

1. $\xymatrix{ \mathbb{Q}^3 \ar[r]^{f_1} & \mathbb{Q}^5 \ar[r]^{f_2} & \mathbb{Q}^2 }$ with $f_1:=\begin{pmatrix} 0 & 1 & 1 \\ 0 & 1 & 1 \\ 2 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 1 & 0\end{pmatrix}$ and $f_2:=\begin{pmatrix} 1 & 1 & 0 & -1 & -1 \\ 2 & 2 & 0 & -2 & -2\end{pmatrix}$
2. $\xymatrix{ \mathbb{Z}^3 \ar[r]^{f_1} & \mathbb{Z}^5 \ar[r]^{f_2} & \mathbb{Z}^2 }$ with $f_1:=\begin{pmatrix} 0 & 1 & 1 \\ 0 & 1 & 1 \\ 2 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 1 & 0\end{pmatrix}$ and $f_2:=\begin{pmatrix} 1 & 1 & 0 & -1 & -1 \\ 2 & 2 & 0 & -2 & -2\end{pmatrix}$

Part B

Let:

$\xymatrix{ M_1 \ar[d]_{f_1} \ar[r] & M_2 \ar[r] \ar[d]_{f_2} & M_3 \ar[r] \ar[d]_{f_3} & M_4 \ar[r] \ar[d]_{f_4} & M_5 \ar[d]_{f_5} \\ N_1 \ar[r] & N_2 \ar[r] & N_3 \ar[r] & N_4 \ar[r] & N_5 }$

be a commutative diagram of $R$-modules in which the rows are exact sequences. Show the Five lemma:

• If $f_1,f_2,f_4$ and $f_5$ are isomorphism then so is $f_3$

Problem 3

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TODO: Problem statement

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Solution

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TODO: Solution

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Problem 4

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TODO: Problem statement

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Solution

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TODO: Solution

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