What is the submodule of $\prod_{p}{\mathbb{Z}/(p)}$ viewed as a $\mathbb{Z}$-module. In particular,what about its maximal submodule? I feel that the maximal submodule may like this: $$\{(a_p)_p\in\prod_{p}{\mathbb{Z}/(p)}|a_q=0~in~\mathbb{Z}/(p)\},$$ for some prime $q$.
Maximal submodule of the direct product of all prime number order group
0
$\begingroup$
abstract-algebra
group-theory
-
0What do you mean by <
> maximal submodule? The group $\prod_p \mathbf{Z}/p\mathbf{Z}$ has a lot of maximal subgroups. For each prime $q$, the example you provided in your question is a maximal subgroup of index $q$. – 2017-01-03 -
0I just want to say that all the maximal submodule of $\prod_p\mathbb{Z}/p\mathbb{Z}$ is this form. But I don't know whether it is right. – 2017-01-03
1 Answers
2
A $\mathbb{Z}$-module is simple iff it has prime order $q$. And a submodule $M$ of $N = \prod_{p}{\mathbb{Z}/(p)}$ is maximal in $N$ iff $N/M$ is simple.
So if $M$ is maximal in $N$, and $N/M$ has prime order $q$, for all $(a_p)_{p}$ you will have $q (a_p)_{p} = (q a_p)_{p} \in M$. Now note that if $p \ne q$ we have that $q \mathbb{Z}/(p) = \mathbb{Z}/(p)$, as $q$ is invertible modulo $p$.