nLab bimodule

Idea

In algebra, a bimodule over a ring is a module in two compatible ways, with one action from the left and one from the right.

This notion generalizes to the broader context of actions of (enriched) categories, in which case bimodules are also called profunctors or distributors.

Definition

Over a ring

With a left action and a right action

Given two rings $R$ and $S$, a $R$-$S$-bimodule is an abelian group $B$ with a bilinear left $R$-action $\alpha_R:R \times B \to B$ and a bilinear right $S$-action $\alpha_S:B \times S \to B$ such that for all $r \in R$, $b \in B$, and $s \in S$, $\alpha_R(r, \alpha_S(b, s)) = \alpha_S(\alpha_R(r, b), s)$.

With a biaction

Equivalently, given two rings $R$ and $S$, a $R$-$S$-bimodule is an abelian group $B$ with a trilinear $R$-$S$-biaction, a function $(-)(-)(-):R \times B \times S \to B$ such that

• for all $b \in B$, $1_R b 1_S = b$

• for all $b \in B$, $r_1 \in R$, $r_2 \in R$, $s_1 \in S$, $s_2 \in S$, $r_1 (r_2 b s_1) s_2 = (r_1 \cdot_R r_2) b (s_1 \cdot_S s_2)$

• for all $r_1 \in R$, $r_2 \in R$, $b \in B$, $s \in S$, $(r_1 + r_2) b s = r_1 b s + r_2 b s$

• for all $r \in R$, $b_1 \in B$, $b_2 \in B$, $s \in S$, $r (b_1 + b_2) s = r b_1 s + r b_2 s$

• for all $r \in R$, $b \in B$, $s_1 \in S$, $s_2 \in S$, $r b (s_1 + s_2) = r b s_1 + r b s_2$

representing simultaneous left multiplication by scalars $r \in R$ and right multiplication by scalars $s \in S$.

Over a fixed ring

In the case $R=S$ the category of $(R,R)$-bimodules can be described as the category of abelian group objects in the slice category of (not necessarily commutative) rings over $R$. See Beck module for a proof.

Over a monoid in a monoidal category

We can define in more generality what is a $(A,B)$-bimodule in a monoidal category $(\mathcal{C},\otimes,I)$ where $(A,\nabla^{A},\eta^{A})$ and $(B,\nabla^{B},\eta^{B})$ are two monoids. It is given by:

• An object $X \in \mathcal{C}$
• A left-action $l:A \otimes X \rightarrow X$
• A right-action $r:X \otimes B \rightarrow X$

such that:

• $(X,l)$ is a left module
• $(X,r)$ is a right module

and moreover this diagram commutes:

Properties

Biactions, left actions, and right actions

Let $R$ and $S$ be rings, and let $B$ be a $R$-$S$-bimodule.

Given a left $R$-action $\alpha_R$ and a right $S$-action $\alpha_S$ of a $R$-$S$-bimodule, the biaction $(-)(-)(-):R \times B \times S \to B$ is defined as

$$r b s \coloneqq \alpha_R(r, \alpha_S(b, s)) = \alpha_S(\alpha_R(r, b), s)$$

The biaction is trilinear because the left $R$-action and right $S$-action are bilinear.

On the other hand, given an $R$-$S$-biaction $\alpha$ of a $R$-$S$-bimodule, the left $R$-action is defined from the $R$-$S$-biaction as

$$\alpha_R(r, b) \coloneqq r b 1_S$$

for all $r \in R$ and $b \in B$. It is a left action because

$$\alpha_R(1_R, b) = 1_R b 1_S = m$$

$$\alpha_R(r_1, \alpha_L(r_2, b)) = r_1 (r_2 b 1_S) 1_S = (r_1 \cdot_R r_2) b (1_S \cdot_S 1_S) = (r_1 \cdot_R r_2) b 1_S = \alpha_R(r_1 \cdot_R r_2, b)$$

The right $S$-action is defined from the $R$-$S$-biaction as

$$\alpha_S(b, s) \coloneqq 1_R b s$$

for all $s \in S$ and $b \in B$. It is a right action because

$$\alpha_S(b, 1_S) = 1_R b 1_S = m$$

$$\alpha_S(\alpha_S(b, s_1), s_2) = 1_R (1_R, b, s_1) s_2 = (1_R \cdot_R 1_R) b (s_1 \cdot_S s_2) = 1_S b (s_1 \cdot_S s_2) = \alpha_S(b, s_1 \cdot_S s_2)$$

The left $R$-action and right $S$-action satisfy the following identity:

• for all $b \in B$, $r \in R$ and $s \in S$, $\alpha_R(r, \alpha_S(b, s)) = \alpha_S(\alpha_R(r, b), s)$.

This is because when expanded out, the identity becomes:

$$\alpha(r, \alpha(1_R, b, s), 1_S) = \alpha(1_R, \alpha(r, b, 1_S), s)$$

$$(r \cdot_R 1_R) b (s \cdot_S 1_S) = (1_R \cdot_R r) b (1_S \cdot_S s)$$

$$r b s = r b s$$

The left $R$-action and right $S$-action are bilinear because the original biaction is trilinear.

Linear maps

Let $R$ and $S$ be rings. A $R$-$S$-linear map or $R$-$S$-bimodule homomorphism between two $R$-$S$-bimodules $A$ and $B$ is an abelian group homomorphism $f:A \to B$ such that for all $a \in A$, $r \in R$, and $s \in S$,

$$f(r a s) = r f(a) s$$

A $R$-$S$-linear map $f:A \to B$ is monic or an $R$-$S$-bimodule monomorphism if for every other $R$-$S$-bimodule $C$ and $R$-$S$-linear maps $h:C \to A$ and $k:C \to A$, $f \circ h = f \circ k$ implies that $h = k$.

A sub-$R$-$S$-bimodule of a $R$-$S$-bimodule $B$ is a $R$-$S$-bimodule $A$ with a monic linear map $i:A \hookrightarrow B$.

A $R$-$S$-linear map $f:A \to B$ is invertible or an $R$-$S$-bimodule isomorphism if there exists a $R$-$S$-linear map $g:B \to A$ such that $g \circ f = id_A$ and $f \circ g = id_B$, where $id_A$ and $id_B$ are the identity linear maps on $A$ and $B$ respectively.

Tensor product of bimodules

Given rings $R, S, T$ and an $R$-$S$ bimodule $A$ and an $S$-$T$ bimodule $B$, the tensor product of $A$ and $B$ is formed as a quotient $A \otimes_S B$ of the tensor product of abelian groups $A\otimes B$. This is a special case of a more general construction:

Given three monoids $M,N,P$ in a monoidal category $(\mathcal{C},\otimes,I)$, an $M$-$N$-bimodule $A$ and an $N$-$P$-bimodule $B$, we denote the monoid actions as $\lambda^{A}:M \otimes A \rightarrow A$, $\rho^{A}:A \otimes N \rightarrow A$, $\lambda^{B}:N \otimes B \rightarrow B$ and $\rho^{B}:B \otimes P \rightarrow B$. The tensor product, $A \otimes_{N} B$ is defined as this coequalizer:

We suppose moreover that this coequalizer is preserved by tensoring on the left by $M$ and tensoring on the right by $P$, meaning that these diagrams are coequalizer diagrams: $A \otimes_{N} B$ becomes an $M$-$P$-bimodule with left action defined by the following diagram: and right action defined by the following diagram:

Assuming all requisite (reflective) coequalizers exist, universal property arguments guarantee associativity isomorphisms of type

$$(A \otimes_N B) \otimes_P C \to A \otimes_N (B \otimes_P C).$$

In fact, this tensor product defines composition in a bicategory where objects or 0-cells are monoids in a monoidal category, where 1-cells $A$ from $R$ to $S$ are $R$-$S$ bimodules, and where 2-cells from $A$ to $B$ are morphisms of $R$-$S$ bimodules.

This in turn can be seen as a special case of a bicategory of profunctors enriched in a monoidal category with suitably nice cocompleteness properties – see monoidally cocomplete category and Benabou cosmos.

Two-sided ideals of a ring

Every ring $R$ is a $R$-$R$-bimodule, with the biaction $(-)(-)(-):R \times R \times R \to R$ defined by the ternary product $a b c \coloneqq a \cdot b \cdot c$ for elements $a \in R$, $b \in R$, $c \in R$.

Given a ring $R$, a two-sided ideal of $R$ is a sub-$R$-$R$-bimodule of $R$.

Rings over a ring

Let $R$ be a ring. An $R$-ring $S$ is a $R$-$R$-bimodule with a bilinear function $(-)\cdot(-):S \times S \to S$ and an element $1 \in S$ such that $(S, \cdot, 1)$ forms a monoid.

Categories of bimodules

The 1-category of bimodules and intertwiners

Definition

Write $BMod$ for the category whose

• objects are triples $(R,S,B)$ where $R$ and $S$ are rings and where $B$ is an $R$-$S$-bimodule;

• morphisms are triples $(f,g, \phi)$ consisting of two ring homomorphisms $f \colon R \to R'$ and $g \colon S \to S'$ and an intertwiner of $R$-$S'$-bimodules $\phi \colon B \cdot g \to f \cdot B'$. This we may depict as a

  $$\array{ R &\stackrel{B}{\to}& S \\ {}^{\mathllap{f}}\downarrow &\Downarrow_{\phi}& \downarrow^{\mathrlap{g}} \\ R' &\stackrel{B'}{\to}& S' } \,.$$

Remark

As this notation suggests, $BMod$ is naturally the vertical category of a pseudo double category whose horizontal composition is given by tensor product of bimodules.

The 2-category of rings, bimodules, and intertwiners

Consider bimodules over rings.

Proposition

There is a 2-category whose

• objects are rings;

• 1-morphisms are bimodules;

• 2-morphisms are intertwiners.

The composition of 1-morphisms is given by the tensor product of modules over the middle algebra.

Proposition

There is a 2-functor from the above 2-category of rings and bimodules to Cat which

• sends an ring $R$ to its category of modules $Mod_R$;

• sends a $R$-$S$-bimodule $B$ to the tensor product functor

  $$(-)\otimes_{R} B \;\colon\; Mod_{R} \to Mod_{S}$$

• sends an intertwiner to the evident natural transformation of the above functors.

Proposition

This construction has as its image precisely the colimit-preserving functors between categories of modules.

This is the Eilenberg-Watts theorem.

Remark

In the context of higher category theory/higher algebra one may interpret this as says that the 2-category of those 2-modules over the given ring which are equivalent to a category of modules is that of rings, bimodules and intertwiners. See also at 2-ring.

Remark

The 2-category of rings and bimodules is an archtypical example for a 2-category with proarrow equipment, hence for a pseudo double category with niche-fillers. Or in the language of internal (infinity,1)-category-theory: it naturally induces the structure of a simplicial object in the (2,1)-category $Cat$

$$\left( \cdots \stackrel{\to}{\stackrel{\to}{\to}} X_1 \stackrel{\overset{\partial_1}{\to}}{\underset{\partial_0}{\to}} X_0 \right) \in Cat^{\Delta^{op}}$$

which satisfies the Segal conditions. Here

$$X_0 = Ring$$

is the category of rings and homomorphisms between them, while

$$X_1 = BMod$$

is the category of def. , whose objects are pairs consisting of two rings $A$ and $B$ and an $A$-$B$ bimodule $N$ between them, and whose morphisms are pairs consisting of two ring homomorphisms $f \colon A \to A'$ and $g \colon B \to B'$ and an intertwiner $N \cdot (g) \to (f) \cdot N'$.

The $(\infty,2)$-category of $\infty$-algebras and $\infty$-bimodules

The above has a generalization to (infinity,1)-bimodules. See there for more.

Related concepts

• module over a monad

• bimodule object

• quotient bimodule

• Hilbert bimodule

• Noetherian bimodule

• Artinian bimodule

• Soergel bimodule

• two-sided ideal

• category of two-sided ideals in a ring

• biaction

• amplimorphism

• bibundle

• (infinity,1)-bimodule

• bimodule category

• 2-module, 2-ring

References

The 2-category of bimodules in its incarnation as a 2-category with proarrow equipment appears as example 2.3 in

• Michael Shulman, Framed bicategories and monoidal fibrations (arXiv:0706.1286)

Bimodules in homotopy theory/higher algebra are discussed in section 4.3 of

• Jacob Lurie, Higher Algebra

For more on that see at (∞,1)-bimodule.