course |
course_year |
question_number |
tags |
title |
year |
Groups, Rings and Modules |
IB |
34 |
IB |
2014 |
Groups, Rings and Modules |
|
Paper 3, Section II, E |
2014 |
Let $R$ be a ring, $M$ an $R$-module and $S=\left{m_{1}, \ldots, m_{k}\right}$ a subset of $M$. Define what it means to say $S$ spans $M$. Define what it means to say $S$ is an independent set.
We say $S$ is a basis for $M$ if $S$ spans $M$ and $S$ is an independent set. Prove that the following two statements are equivalent.
-
$S$ is a basis for $M$.
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Every element of $M$ is uniquely expressible in the form $r_{1} m_{1}+\cdots+r_{k} m_{k}$ for some $r_{1}, \ldots, r_{k} \in R$.
We say $S$ generates $M$ freely if $S$ spans $M$ and any map $\Phi: S \rightarrow N$, where $N$ is an $R$-module, can be extended to an $R$-module homomorphism $\Theta: M \rightarrow N$. Prove that $S$ generates $M$ freely if and only if $S$ is a basis for $M$.
Let $M$ be an $R$-module. Are the following statements true or false? Give reasons.
(i) If $S$ spans $M$ then $S$ necessarily contains an independent spanning set for $M$.
(ii) If $S$ is an independent subset of $M$ then $S$ can always be extended to a basis for $M$.