I haven't done any mathematics for a long time, and I have forgotten some things. I want to try to remember some of the words and how they interact.
A module is a 'vectorspace over a ring' rather than over a field. So it is an abelian group under addition, closed under left multiplication from a ring. 1) i) A ring is closed under left multiplication and is also an abelian group, then a ring is itself a module over itself? ii) A field is closed under left multiplication from a field, so a field is itself a 1-dimensional vectorspace over itself.
2) An $R$-algebra is a module with a multiplication, and an algebra is a vectorspace with a multiplication. i) I could let a ring $R$ have multiplication being standard multiplication, meaning a ring $R$ is itself an $R$-algebra, ii) I could left a field $\mathbb F$, which is a vectorspace by 1) i), have multiplication be field multiplication, meaning a field is itself an algebra?
Yes, a ring is a module over itself. The submodules correspond to ideals.
Yes, a field is a 1-dimensional vector space over itself.
For algebras over a ring $R$, we usually assume $R$ to be commutative. Then yes, every commutative ring is an algebra over itself. (More generally, any ring is an algebra over any one of its subrings that is contained in the center.)
And yes, by the same reasoning, any field is an algebra over itself (or any of its subfields).