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?