# Morita contexts without tears

My question is: Has anybody seen Morita contexts introduced as it is done below?

I first intended this as an answer to the question "Reference request: Morita contexts" by Bey,
but then decided to formulate it as a question, since this way it has better chance of being noticed,

Bey gives two references, Rowen's Graduate Algebra: Noncommutative View and McConnell and Robson's Noncommutative Noetherian Rings, complaining that neither spends too much time on Morita contexts, and asks for references to texts that treat this topic in more detail. The comments and an answer give three references: Rowen's Ring Theory - Volume I, Lam's Lectures on modules and rings, and Anderson and Fuller's Rings and categories of modules. Let me add another reference, Jacobson's Basic Algebra II.

The definitions of Morita contexts in the cited texts are anything but illuminating. For example,
the description of the Morita context of a right module in Lam's Lectures on modules and rings (implicitly) involves checking quite a few of "associativity laws", and once you are through with the definition you are left with the uneasy feeling that checking was not thorough enough, that some of the associativity laws were left out. And even when you figure out an exhaustive collection of associativity laws and check them all, you may still wonder what mathemagic is making so many checks succeed.

There is a way of introducing the Morita context of a right module $M=M_R$ over a ring $R$ that does not require any checking of "associativity laws" since the associativity is hardwired into the construction of the context.

As usual, $R_R$ denotes the ring $R$ regarded as the right regular $R$-module. $\newcommand{\r}{\mathrm{r}}\newcommand{\s}{\mathrm{s}}$ Let $\r$ and $\s$ be two distinct objects, let $X\r:=R_R$ and $X\s:=M_R$, $\newcommand{\Hom}{\mathrm{Hom}}\newcommand{\set}{\{#1\}}$ and set $\hom(a,b):=\Hom(Xa,Xb)$ for $a,b\in\set{\r,\s}$. We obtained a preadditive category $A$ on the two objects $\r$ and $\s$. $\newcommand{\End}{\mathrm{End}}$ Let us write $\overline{R}:=\hom(\r,\r)=\End(R_R)$ and $\overline{M}:=\hom(\r,\s)=\Hom(R_R,M_R)$; $\overline{R}$ is a ring and $\overline{M}$ is
an additive group. We define two maps: $\varrho\colon\overline{R}\to R : \varphi\mapsto \varphi(1)$ and $\mu\colon\overline{M}\to M : \psi\mapsto\psi(1)$.
The map $\varrho$ is an isomorphism of rings, the map $\mu$ is an isomorphism of additive groups, $\newcommand{\compose}{\circ}$
and $\mu(\psi\compose\varphi)=(\mu\psi)\cdot(\varrho\varphi)$ for all $\varphi\in\overline{R}$ and all $\psi\in\overline{M}$; that is, $(\mu,\varrho)$ is an isomorphism $\overline{M}_{\overline{R}}\to M_R$ of right modules.

Now we identify $\overline{R}$ with $R$ via $\varrho$, and $\overline{M}$ with $M$ via $\mu$. Using these two identifications, we replace the hom-set $\hom_A(\r,\r)=\overline{R}$ of the category $A$ with the ring $R$, the hom-set $\hom_A(\r,\s)=\overline{M}$ with the module $M$, and leave the other two hom-sets as they are. In this way we obtain the preadditive category $C$ with the hom-sets $$\begin{gathered} \hom_C(\r,\r)=R~, \qquad \hom_C(\r,\s)=M~, \\ S:=\hom_C(\s,\s)=\hom_A(\s,\s)=\End(M_R)~, \\ N:=\hom_C(\s,\r)=\hom_A(\s,\r)=\Hom(M_R,R_R)~; \end{gathered}$$ This category $C$ is the Morita context of the right module $M_R$ $\newcommand{\Mc}{\mathrm{Mc}}$ and shall be denoted $\Mc(M_R)$.

We are going to figure out how the morphisms of the preadditive category $\Mc(M_R)$ are composed. To do this we have to know the inverses of the isomorphisms $\varrho$ and $\mu$. For every $r\in R$ the inverse image $r^*:=\varrho^{-1}r\in\overline{R}$ is the left multiplication by $r$: $~r^*(r')=r\cdot r'$ for $r'\in R$. Similarly for every $m\in M$ the inverse image $m^*:=\mu^{-1}m\in\overline{M}$ is the left multiplication by $m$: $~m^*(r')=m\cdot r'$ for $r'\in R$. $\newcommand{\dia}{\diamond}$ Let us write the composition in $\Mc(M_R)$ as $u\dia v$, to distinguish it from the composition $x\compose y$ in $A$ (which is the composition of functions). Then, for any $r,r_1\in R$, $m\in M$, $s,s_1\in S$,
and $n\in N$, we have $\newcommand{\Eq}{\,=\,}$ \begin{aligned} r\dia r_1 ~&\Eq (r^*\compose r_1^*)(1) \Eq r\cdot r_1~, \\ m\dia r ~&\Eq (m^*\compose r^*)(1) \Eq m\cdot r~, \\ s\dia m ~&\Eq (s\compose m^*)(1) \Eq s(m)~, \\ n\dia m ~&\Eq (n\compose m^*)(1) \Eq n(m)~, \\ s\dia s_1 ~&\Eq s\compose s_1~, \\ n\dia s ~&\Eq n\compose s~, \\ r\dia n ~&\Eq r^*\compose n~, \\ m\dia n ~&\Eq m^*\compose n~. \end{aligned} The last two composites are maps $M\to R$ resp. $M\to M$, where \begin{aligned} (r\dia n)(m') ~&\Eq r\cdot n(m')~, \\ (m\dia n)(m') ~&\Eq m\cdot n(m') \end{aligned} for every $m'\in M$.

Compare the above with the definition of the Morita context of a right module in Lam's Lectures on modules and rings: Lam in effect first introduces the eight species of composites $u\dia v$ above, then starts (but only starts) checking the associativity of this composition. All in all there are sixteen cases to check, all checks are straightforward --- but, as we see now, completely unnecessary. With our approach the associativity of the composition is a given.

There is also the notion of a Morita context (mark the indefinite article). If you contemplate,
for a moment or two, the definition of a Morita context, say in Rowen's Ring Theory - Volume I,
you will notice that what it describes is a preadditive category on two objects -- no more, no less.

• "you will notice that what it describes is a pre-additive category on two objects -- no more, no less" Very nice viewpoint!! – darij grinberg Oct 6 '14 at 16:39

## 1 Answer

Yes, there is a paper about a generalization of Morita contexts into bicategories. It also contains some references for the special case of rings and bimodules.

A main message of that paper is a similar observation about Morita contexts between categories, say, $\mathcal A$ and $\mathcal B$: these are exactly the categories which disjointly contain (isomorphic copies of) $\mathcal A$ and $\mathcal B$ and have no further objects -- such thing I called a 'bridge'.

Another 'easy' approach is that defining a Morita context involving bimodules ${}_AM_B$ and ${}_BN_A$ is equivalent to defining (the ring operation of) a generalized matrix ring $$\pmatrix{A&M\\N&B}:=\left\{\pmatrix{a&m\\n&b}:a\in A,b\in B,m\in M,n\in N\right\}\,.$$

• Not quite what I was looking for. My aim was to build a clear mental picture of the Morita context, based on as simple and "natural" structure as possible; your paper, deep as it is, offers no insight in this direction. Concerning the generalized matrix ring: it requires, a priory, that $A$, $B$, $M$, $N$ are hom-sets of a pre-additive category on two objects. This ring is in turn a special case of the ring (that has the identity element decomposed into pairwise orthogonal idempotents) associated with a pre-additive category on any finite set of objects. – chizhek Oct 7 '14 at 9:33
• Isn't Thm.5.4. of the flavor that you are looking for? Also, there are some citations for older papers on Morita contexts among rings.. – Berci Oct 7 '14 at 20:32
• Theorem 5.4 indeed has the right flavor. However, the view of a Morita context as a two-object Ab-category is obscured by a reference to a previous, more general development in the paper. A Morita context of a (left/right) module may be just a special case of some more general notion -- every mathematical notion is a special case of some more general notion, perhaps not yet known -- but it has a peculiar life of its own. – chizhek Oct 8 '14 at 13:12