nLab ring over a ring




Recall that AA-algebras, in the usual sense of associative algebras over a commutative ring AA, are rings RR equipped with ring homomorphisms η:AZ(R)\eta \colon A\to Z(R) from AA to their center (and a plain unital ring RR is an associative algebra over the integers \mathbb{Z}).

By the term AA-rings (e.g. Böhm (2009), §2.1) one refers to the generalization of this notion to the case where AA is not necessarily commutative or the image of η\eta is not necessarily in the center of RR. Hence AA-rings are general objects in the coslice category A/RingsA/Rings (also denoted ARingA \downarrow Ring) for arbitrary AA.

The dual notion is, in some sense, that of AA-corings (e.g. Böhm (2009), §2.2).



For AA a (possibly noncommutative) ring, a ring over AA, or AA-ring for short, is a monoid object RR internal to the category A Bimod A Bimod (that is: AMod A{}_A Mod_A), understood as a monoidal category with respect to the tensor product of bimodules.

(e.g. Böhm (2009), Def. 2.1)

Every AA-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 ARA \to R is a morphism of rings, and the category of AA-rings is precisely the coslice category or coslice category A/RingA/Ring. Thus by category-theoretic rules, one might be led to unconventionally call AA-rings “rings under AA”. Unfortunately, standard name for AA-rings is “rings over AA”, like conventionally calling kk-algebras the “algebras over KK”.

Unlike for the kk-algebras, the multiplication R×RRR\times R\to R which is the morphism of AA-bimodules, is not (left) AA-linear in the second factor, but only A opA^{op}-linear (that is, AA-linear on the right). In other words, the axiom for KK-algebras k(rs)=r(ks)k (r s) = r (k s) is not true, for kAk\in A, r,sRr,s\in R, although k(rs)=(kr)sk (r s) = (k r) s and (rs)k=r(sk)(r s) k = r (s k) do hold.

Both for a discussion for under-over and also for this difference between KK-algebras and AA-rings see the Café's quick algebra quiz.

AA opA\otimes A^{op}-rings

The structure of an AA opA\otimes A^{op}-ring (R,μ,η)(R,\mu,\eta) is determined by the structure of AA as a ring, together with the two natural homomorphisms of rings s=η(1 A):ARs = \eta(-\otimes 1_A):A\to R and t=η(1 A):A opRt=\eta(1_A\otimes -):A^{op}\to R which have commuting images (s(a)t(a)=t(a)s(a)s(a)t(a')=t(a')s(a), for all a,aAa,a'\in A). 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 AA-rings, and especially useful in the study of AA opA\otimes A^{op}-rings, given by certain coend, so called Takeuchi product.


Original articles:

Introduction and review:

See also

  • T. Brzeziński, G. Militaru, Bialgebroids, × R\times_{R}-bialgebras and duality, J. Algebra 251: 279-294, 2002, math.QA/0012164

  • P. Schauenburg, Bialgebras over noncommutative rings and a structure theorem for Hopf bimodules, Appl. Categ. Structures 6 (1998), 193–222, ps doi

Last revised on June 13, 2023 at 12:13:02. See the history of this page for a list of all contributions to it.