# Is algebra over a set also algebra over a field?

During my studies I have come across two different notions of the term "algebra", namely algebra over a set and algebra over a field (the field its vector space always being Euclidean space in my case).

While both of those concepts use the same term, I can't seem to find any relation between them, e.g. one being a sub-case of the other. Is there any such relation? Is there a reason why they are called the same?

An algebra over a set is actually a boolean ring: if we interpret $+$ as symmetric difference $\triangle$ and $\cdot$ as intersection $\cap$, they satisfy all (unital) ring axioms (with $0=\emptyset$ and $1$ equal to the universe), and moreover $x+x=x\triangle x=\emptyset=0$. (However, it is never a field, unless it has two elements, despite being called a field of sets: no element except $1$ itself has a $\cap$-inverse.)