To receive notifications about scheduled maintenance, please subscribe to the mailing-list gitlab-operations@sympa.ethz.ch. You can subscribe to the mailing-list at https://sympa.ethz.ch

Unverified Commit f17a74ef authored by rrueger's avatar rrueger
Browse files

Algebra I: Modules. Update definition of torsion

parent f2f977e0
......@@ -219,14 +219,15 @@ $K$. Then we have $M \cong D^{(n)}/K$. The core idea is now to examine $K$.
\end{remark}
\begin{definition}
Let $M$ be a finitely generated module over a PID and define $\tor M $ by
Let $M$ be a finitely generated module over a PID and define the
\textbf{torsion module} of $M$, $\tor M$, by
%
\begin{align*}
\tor M = \set{y \in M \mid \exists 0 \neq a \in D: ay = 0}.
\tor M = \set{y \in M \mid \exists a \in D \setminus \set{0}: ay = 0}.
\end{align*}
%
Then $y \in \tor M$ if and only if $\ann y \neq 0$ and $\tor M$ is a submodule
of $M$. We call this the \textbf{torsion module} of $M$.
Then $y \in \tor M$ if and only if $\ann y \neq \set{0}$ and $\tor M$ is a
submodule of $M$.
\end{definition}
\begin{theorem}
......
Markdown is supported
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment