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 $A$. But that is effectively just the algebra over the center $Z(R)$, so we do not need a new concept(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: $Z(R)$-algebras. 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$-coalgebra to a noncommutative ground ring. Edit: I suggest also this link. 2 added 291 characters in body The commutative notion of an (associative or not) algebra$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$. But that is effectively just the algebra over the center$Z(R)$, so we do not need a new concept. Thus the two useful concepts in the noncommutative case are$R$-rings (possibly nonassociative!) and, well,$Z(R)$-algebras. There is also a notion of$R$-coring, which is a comonoid in the monoidal category of$R$-bimodules, generalizing the notion of an$R$-coalgebra to a noncommutative ground ring. 1 The commutative notion of an associative algebra$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$which is left linear in first and right linear in the second factor. In the commutative case, the mutliplication is both left and right linear in each factor, what is here possible only if$R$maps into the center of$A$. But that is effectively just the algebra over the center$Z(R)$, so we do not need a new concept. Thus the two useful concepts in the noncommutative case are$R$-rings and, well,$Z(R)$-algebras. There is also a notion of$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.