2
$\begingroup$

Suppose $R=\mathbb{Z}/4\mathbb{Z}$.

i) How many R-submodules $M= Rx \subset R^{2} \ (x\in R^{2}) $ are there?
ii) How many equivalence classes are isomorphic to M?

i) definition of a submodule: Let M be a R-module an $L\subset M$, then L is a R-submodule of M if L itself is a R-module respectively to the operation on M.

every ideal I of R is a R-submodule of R. The ideals of R are the equivalence classes $\overline{0,1,2,3}$, so the submodules of M in this case are : 0x,1x,2x,3x. From this it would also follow that the number of equivalence classes is equal to the number of the submodules (ii).

Is this reasoning alright?

  • 0
    I don't really understand the second question at all. Are you just asking for the isomorphism classes of these $M$ that we got in the first part?2011-12-01
  • 0
    @ Dylan Moreland: Thank you... I'm not sure myself. This is an example from the text: $M=\mathbb{Z}^{3}, L=\mathbb{Z}(1,2,3) + \mathbb{Z}(4,5,6) + \mathbb{Z}(7,8,9), x_{1} = (1,2,3), t_{1}=1, M'= \{(0,y,z);y,z\in \mathbb{Z}\}$ $\Rightarrow L'=\mathbb{Z}(0,3,6)+\mathbb{Z}(0,6,12)=\mathbb{Z}(0,3,6)$ $\Rightarrow$ the classes in M/L are: $a(0,1,2)+bx_{3}+L , 0\le a < 3 , b\in \mathbb{Z}$. $\Rightarrow M/L \approx (\mathbb{Z}/3 \mathbb{Z})\times \mathbb{Z}$2011-12-01
  • 0
    Hm. I don't really follow the example. What is the text, out of curiosity?2011-12-02

1 Answers 1

2

$\newcommand{\Z}{\mathbf{Z}}$Let's make sure we know how to deal with $R$, first! Maybe it's helpful to note that we're really asking questions about abelian groups here: an $R$-submodule of $R$ is the same as a $\Z$-submodule of $R$. In any event, the submodules of $R$ correspond to subgroups of $\Z$ which contain $4\Z$, and there are three of these.

Now you want to find the cyclic $R$-submodules of $R \times R$. This module only has $16$ elements, so it isn't that hard to just check each one [Say I take $(1, 2) \in R \times R$. Then this generates $\{(1, 2), (2, 0), (3, 2), (0, 0)\}$, which is isomorphic to $R$]. But some observations can make the job easier.

Note that the order of $(a, b)$ is the l.c.m. of the orders of $a$ and $b$. If I want to find the cyclic submodules of order $4$, i.e. the submodules isomorphic to $R$, then I want $a$ or $b$ of order $4$. So I want pairs like $(1, b), (3, b)$, and the mirror versions of these. By my count, there are $12$ such pairs. Here we have to be careful: each of these elements will generate a submodule that contains two elements of order $4$, so in the end there are only $6$ submodules of this sort.