TITLE: Definition of an algebra over a noncommutative ring
QUESTION [26 upvotes]: 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?

REPLY [3 votes]: This is a follow-up to the answer by Zoran Skoda, applicable to the case where one requires "algebras" to be associative and have a unit.
I'd like to point out that a monoid $(A,\mu,\eta)$ internal to the monoidal category of $R$-bimodules is exactly the same thing as a ring $A$ along with a ring homomorphism $\eta:R\rightarrow A$. And a morphism $(A,\mu,\eta)\rightarrow (A',\mu',\eta')$ is exactly the same thing as a ring morphism $A\rightarrow A'$ that makes the triangle with $R$ commute.
In other words, this viewpoint can be greatly simplified if we just view an "algebra over $R$" as an object $\eta:R\rightarrow A$ in the category of rings under $R$. One can require that $\eta:R\rightarrow A$ have central image, or not, depending on one's needs.