symmetric monoidal (∞,1)-category of spectra
Recall that -algebras, in the usual sense of associative algebras over a commutative ring , are rings equipped with ring homomorphisms from to their center (and a plain unital ring is an associative algebra over the integers ).
By the term -rings (e.g. Böhm (2009), §2.1) one refers to the generalization of this notion to the case where is not necessarily commutative or the image of is not necessarily in the center of . Hence -rings are general objects in the coslice category (also denoted ) for arbitrary .
The dual notion is, in some sense, that of -corings (e.g. Böhm (2009), §2.2).
For a (possibly noncommutative) ring, a ring over , or -ring for short, is a monoid object internal to the category (that is: ), understood as a monoidal category with respect to the tensor product of bimodules.
Every -ring is a ring in the usual sense, in the sense that there is an obvious forgetful functor to Ring. In fact the unit map is a morphism of rings, and the category of -rings is precisely the coslice category or coslice category . Thus by category-theoretic rules, one might be led to unconventionally call -rings “rings under ”. Unfortunately, standard name for -rings is “rings over ”, like conventionally calling -algebras the “algebras over ”.
Unlike for the -algebras, the multiplication which is the morphism of -bimodules, is not (left) -linear in the second factor, but only -linear (that is, -linear on the right). In other words, the axiom for -algebras is not true, for , , although and do hold.
Both for a discussion for under-over and also for this difference between -algebras and -rings see the Café's quick algebra quiz.
The structure of an -ring is determined by the structure of as a ring, together with the two natural homomorphisms of rings and which have commuting images (, for all ). In theory of associative bialgebroids these are the dual versions of the source and target maps from the study of groupoids.
There is (not always associative) product of -rings, and especially useful in the study of -rings, given by certain coend, so called Takeuchi product.
Original articles:
Introduction and review:
See also
T. Brzeziński, G. Militaru, Bialgebroids, -bialgebras and duality, J. Algebra 251: 279-294, 2002, math.QA/0012164
Peter Schauenburg, Bialgebras over noncommutative rings and a structure theorem for Hopf bimodules, Appl. Categ. Structures 6 (1998) 193–222, ps doi
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