TITLE: Does Morita theory hint higher modules for noncommutative ring?
QUESTION [11 upvotes]: Two possibly noncommutative rings are called Morita equivalent if their left-module categories are equivalent. In the commutative case, Morita equivalence is nothing more than ring isomorphism. Otherwise, there are many known examples where this does not hold.
That means modules alone are not enough to characterize the ring. Are there notions of higher modules or higher structures, such that the corresponding higher Morita equivalence is nothing more than ring isomorphism?
In short, can you find a better notion of modules that faithfully capture their underlying ring?

REPLY [3 votes]: Although I think my answer "pointed categories" is an important one, there is another way that the question could be interpreted: What is an interesting class of rings which are recoverable up to isomorphism from their categories of modules? And perhaps, if we are not trying to be too universal or functorial, we won't mind if the isomorphism is not unique.
For this, I strongly recommend Morita's original paper introducing his equivalences. (Kiiti Morita, Duality for modules and its applications to the theory of rings with minimum condition, Sci. Rep. Tokyo Kyoiku Daigaku, Sect. A 6, 83-142 (1958). You can find a PDF by googling.) Among the many things in that paper is a theorem saying that rings satisfying a natural minimality condition called "basic" are Morita equivalent iff they are isomorphic (commutative rings, I believe, satisfy this condition) and that every finite-dimensional ring is Morita equivalent to one satisfying this minimality condition. See my answer to a related question.