Definition of an algebra over a noncommutative ring 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?
 A: [Edit: Thanks to Harry Gindi for pointing out that I'm only extending the notion of an associative algebra.]
Here is a somewhat "brute force" approach to define an associative algebra over a general ring.  Let's say that a ring $A$ with a fixed ring homomorphism $f\colon R\to A$ is centrally generated over $R$ (with respect to $f$) if $A$ is generated as a ring by the image of $R$ and a subset $X\subseteq A$ such that every element of the image of $R$ commutes with every element of $X$.  
Then it's clear that whenever $R$ is commutative, a ring homomorphism $f\colon R\to A$ makes $A$ into an $R$-algebra if and only if $A$ is centrally generated over $R$ with respect to $f$.
A: Why not just: $A$ is a ring together with a ring homomorphism $R\to A$?
A: 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.
A: The commutative notion of an (associative or not) algebra $A$ over a commutative ring $R$ has two natural generalization to the noncommutative setup, but the one you list with defined left $R$-linearity in both arguments is neither of them; in particular your multiplication does not necessarily induce a map from the tensor product, unless the image of $R$ is in the center. Most useful is the notion of an $R$-ring $A$ (or a ring $A$ over $R$), which is just a monoid in the monoidal category of $R$-bimodules: in other words the multiplication is a map $A\otimes A\to A$ which is left linear in first and right linear in the second factor. If we drop the associativity for the multiplication all works the same way, but I do not know if there is a common name (maybe descriptive like magma internal to the monoidal category of $R$-bimodules; or one may try a rare term nonassociative $R$-ring). 
In the commutative case, the mutliplication is both left and right linear in each factor, what is here possible only if $R$ maps into the center of $A$. (Edit: I erased here one  additional nonsense sentence clearly written when tired ;) ). Thus the two useful concepts in the noncommutative case are $R$-rings (possibly nonassociative!) and, well, the subclass with that property: $R$ maps into $Z(A)$, deserving the full name of "algebra". There is also a notion of $R$-coring, which is a comonoid in the monoidal category of $R$-bimodules, generalizing the notion of an $R$-coalgebra to a noncommutative ground ring.
Edit: I suggest also this link.
A: Unfortunately that's not exactly what you want but in this paper the authors define Lie algebras over noncommutative rings.
A: I would argue that this notion doesn't have one natural generalization.  One obvious one is what I would call an $R$-bimodule algebra, that is, an algebra which is an R-bimodule in such a way that left multiplication of $A$ commutes with right multiplication of $R$ and vice versa.  If you only have one of these actions, you would have an R-left or right module algebra.
On some level, you can't really expect there to be one correct generalization; which one is right depends on the context.  If you have an example, pick the definition that fits your example, and if you don't have an example, why worry? 
