extension of derivation of algebras

Question

I am studying extension of derivations, but I am confused of some notations and maybe symbols! For some more details, I recall a theorem. We have the following theorem :

Theorem: Let $A$ be an algebra separable over its center $C$ and $M$ be an $A \otimes A^{op}$-module. Then any derivation $d: C \to M^{A}$ extends to $\bar{d}: A \to M$.

My questions are as follows:

• What does it mean: Algebra over its center? Is it possible to consider Algebra over any field?
• What is $A \otimes A^{op}$-module? I need more details on $ A^{op}$?
• Can we consider $d$ as a derivation from subalgebra of $A$ instead of $C$?
• What does this symbol $M^{A}$ stand for?

Answer 1

• For any field $k$, a $k$-algebra (associative with unit) is a ring $A$ together with a ring homomorphism $k\ \longrightarrow\ C$, where $C$ is the center of $A$. If the center $C$ is a field, then $A$ is also a $C$-algebra because the inclusion $C\ \longrightarrow\ A$ is a ring homomorphism.
• The structure of $A$ tells you everything about $A^{\text{op}}$; they are isomorphic as rings.
• Yes, because $C$ is a subalgebra of $A$.
• Thanks to Mariano Suárez-Alvarez: The symbol $M^A$ denotes the subspace of $M$ of central elements, i.e. elements $m\in M$ for which $am=ma$ holds for all $a\in A$.