I wonder if there is a name for:
1) Algebras which are Morita equivalent to their centers, or
2) dg-algebras which are derived Morita equivalent to their Hochshild cohomology?
For instance, (finite-dimensional?) semi-simple Frobenius algebras satisfy (1), as often do the smash products of Hopf algebras with commutative algebras. Item (1) also seems closely related to the defining property of an Azumaya algebra. I am less familiar with examples of (2).
The request for a name is just to help me search for examples, classification tools, and intuition in the literature, as (1) and (2) as I wrote them contain too many buzzwords for me to search effectively, it seems.
Anyone who can offer such insight directly will of course be greatly appreciated!