I would like to know whether the following paragraph regarding right ideals and modules is correct. Any comment or help is welcome:
A right ideal of $R$ is just a submodule of the right $R$-module $R_R$ using the operation in $R$. To put it in the conventional symbols: a right ideal $T$ of a ring $(R,+,⋅)$ is a subgroup of $(R,+)$ such that for all $t ∈ T$, $r ∈ R$, we have $t⋅r ∈ T$. The quotient $R_R/T_R$ is itself only a right module: it is a ring only if $T$ is a two-sided ideal.
Given a ring $R$, right $R$-modules are denoted with $R$ as a right index as in the following: $$M_R$$ this notation recalls you that you can multiply elements of $M_R$ with elements of $R$ on the right i.e. multiplication is a map $$M_R \times R \longrightarrow M_R\ \ \ (m,r) \mapsto mr$$
Now, the right $R$-module $R_R$ consists of the additive group $(R,+)$ with right multiplication $$R_R \times R \longrightarrow R_R \ \ \ (a,b) \mapsto ab$$ elements of $R_R$ must be thought as vectors, elements of $R$ must be thought as scalars.