After having read this post on MathOverflow about the etymology of various terms that are used in abstract algebra like groups, rings and fields, I was wondering how and why did the term algebra come to denote specifically algebra over a field or a commutative ring? Why pick the same name as the discipline to denote a very specific algebraic structure?
This question might not be very useful mathematically, but I'd nevertheless like to know the origin of this nomenclature.
A. A. Albert introduced a $K$-algebra $A$ in the sense that $A$ is a $K$-vector space with distributive bilinear product $A\times A\rightarrow A$, $(a,b)\mapsto a\cdot b$. This includes Lie algebras, Jordan algebras, associative algebras, commutative algebras etc. A natural generalization is to replace the field $K$ by a commutative ring $R$ with $1$. For example, one can consider Lie algebras over the ring $\mathbb{Z}$ (although the name Lie ring is used, too).
The same applies to groups, say $GL_n(K)$ over a field $K$. It can be generalized to $GL_n(R)$ over a commutative ring $R$, and still is called a group ("why is a group over a ring called a group - and is not a group ring").
