In Iain Gordon's survery article "Symplectic reflection algebras" the concept of formal deformations of algebras over semisimple artinian (not necessarily commutative) rings is summarized (chapter 2). Unfortunately, deformations are needed in this generality and there are a few general things I don't understand:

An algebra over a semisimple artinian $\mathbb{C}$-algebra $k$ is "defined" as a $k$-bimodule $A$ with a $k$-bimodule morphism $A \otimes_k A \rightarrow A$. Is this a standard definition and is it correct that there is no associativity or unity assumption? I could not find a single book defining an algebra over a not necessarily commutative ring.

In the definition of a formal deformation given in the survey there also seems to be no associativity or unity assumption. Is this a standard definition and does Hochschild cohomology also work in this setting with the same interpretation? My problem is that when I don't restrict my deformations to be associative or have a unit, I might get a lot more deformations.

Is there some literature discussing this in more detail?