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?



Definition of an algebra over a noncommutative ring
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](http://ncatlab.org/nlab/show/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](http://golem.ph.utexas.edu/category/2008/12/a_quick_algebra_quiz.html).


