I am trying to answer a couple questions about counting $F[x]$-submodules and I think I am having trouble understanding definitions. I will try to explain how I have been approaching the problem and hope people will point out what I am doing wrong.
Let $F = \mathbb{Z} / p \mathbb{Z}$ be the finite field with $p$ and consider the $F[x]$-module $V = F[x]/(x^2) \oplus F[x]/(x^3)$.
Question 1. How many $F[x]$-submodules with $p$ elements does $V$ have.
I think the answer is 5 submodules which I will list explicitly:
$\{ v \in V : v = (a,0), \,\, a \in \mathbb{Z_p} \}$ $\{ v \in V : v = (0,b), \,\, b \in \mathbb{Z_p} \}$ $\{ v \in V : v = (ax,0), \,\, a \in \mathbb{Z_p} \}$ $\{ v \in V : v = (0,bx), \,\, b \in \mathbb{Z_p} \}$ $\{ v \in V : v = (0,bx^2), \,\, b \in \mathbb{Z_p} \}$
Is this correct or did I miss something?
Question 2. How many cyclic $F[x]$-submodules with $p^2$ elements does $V$ have and how many noncyclic $F[x]$-submodules with $p^2$ elements does $V$ have?
This is where I get confused in constructing modules with $p^2$ elements.
Should I consider sets of the form $\{ v \in V : v = (a+bx,0), \,\, a,b \in \mathbb{Z_p} \}$ will this still be cyclic?
$\{ v \in V : v = (ax,bx^2), \,\, a ,b\in \mathbb{Z_p} \}$ $\{ v \in V : v = (ax,bx), \,\, a,b \in \mathbb{Z_p} \}$ will these have $p^2$ elements and not be cyclic?
Thanks for your help