symmetric monoidal (∞,1)-category of spectra
Recall that $A$-algebras, in the usual sense of associative algebras over a commutative ring $A$, are rings $R$ equipped with ring homomorphisms $\eta \colon A\to Z(R)$ from $A$ to their center (and a plain unital ring $R$ is an associative algebra over the integers $\mathbb{Z}$).
By the term $A$-rings (e.g. Böhm (2009), §2.1) one refers to the generalization of this notion to the case where $A$ is not necessarily commutative or the image of $\eta$ is not necessarily in the center of $R$. Hence $A$-rings are general objects in the coslice category $A/Rings$ (also denoted $A \downarrow Ring$) for arbitrary $A$.
The dual notion is, in some sense, that of $A$-corings (e.g. Böhm (2009), §2.2).
For $A$ a (possibly noncommutative) ring, a ring over $A$, or $A$-ring for short, is a monoid object $R$ internal to the category $A Bimod$ (that is: ${}_A Mod_A$), understood as a monoidal category with respect to the tensor product of bimodules.
Every $A$-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 $A \to R$ is a morphism of rings, and the category of $A$-rings is precisely the coslice category or coslice category $A/Ring$. Thus by category-theoretic rules, one might be led to unconventionally call $A$-rings “rings under $A$”. Unfortunately, standard name for $A$-rings is “rings over $A$”, like conventionally calling $k$-algebras the “algebras over $K$”.
Unlike for the $k$-algebras, the multiplication $R\times R\to R$ which is the morphism of $A$-bimodules, is not (left) $A$-linear in the second factor, but only $A^{op}$-linear (that is, $A$-linear on the right). In other words, the axiom for $K$-algebras $k (r s) = r (k s)$ is not true, for $k\in A$, $r,s\in R$, although $k (r s) = (k r) s$ and $(r s) k = r (s k)$ do hold.
Both for a discussion for under-over and also for this difference between $K$-algebras and $A$-rings see the Café's quick algebra quiz.
The structure of an $A\otimes A^{op}$-ring $(R,\mu,\eta)$ is determined by the structure of $A$ as a ring, together with the two natural homomorphisms of rings $s = \eta(-\otimes 1_A):A\to R$ and $t=\eta(1_A\otimes -):A^{op}\to R$ which have commuting images ($s(a)t(a')=t(a')s(a)$, for all $a,a'\in A$). 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 $A$-rings, and especially useful in the study of $A\otimes A^{op}$-rings, given by certain coend, so called Takeuchi product.
Original articles:
Introduction and review:
See also
T. Brzeziński, G. Militaru, Bialgebroids, $\times_{R}$-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.