I've tried in vain to find a definition of an algebra over a noncommutative ring. Does this algebraic structure not exist? In particular, does the following definition from http://en.wikipedia.org/wiki/Algebra_(ring_theory) make sense for noncommutative $R$?
Let $R$ be a commutative ring. An algebra is an $R$-module $A$ together with a binary operation $$ [\cdot,\cdot]: A\times A\to A $$ called $A$-multiplication, which satisfies the following axiom: $$ [a x + b y, z] = a [x, z] + b [y, z], \quad [z, a x + b y] = a[z, x] + b [z, y] $$ for all scalars $a$, $b$ in $R$ and all elements $x$, $y$, $z$ in $A$.
So, is there a common notion of an algebra over a noncommutative ring?