Submodule example

Question

Let $B$ and $C$ be submodules of an $R$-module $A$.

The definition of a submodule $T$ of $A$ is a subgroup $T$ of $A$ so that for all $r \in R$ and for all $t \in T$, $rt \in T$.

Answer 1

For $B\cap C$, I let you manage (and you right it is a module), it's very easy to prove it. For $B\cup C$, it's even not a group a priori ! So it won't be a module. For counter-example just think about two lines over $\mathbb R$ such that the intersection is only $\{0\}$.

Answer 2

If $rx\in B\wedge rx\in C$, why shouldn't $rx\in B\cap C$?

The union of two two dimensional subspaces over $\mathbb R$ is only a subspace of $\mathbb R^3$ if they are equal.