Mathematics

Questions tagged [bimodules]

For questions about bimodules, abelian groups that are both a left and a right module, such that the left and right multiplications are compatible.

In abstract algebra, a bimodule is an abelian group that is both a left and a right module, such that the left and right multiplications are compatible. Besides appearing naturally in many parts of mathematics, bimodules play a clarifying role, in the sense that many of the relationships between left and right modules become simpler when they are expressed in terms of bimodules.

Source: https://en.wikipedia.org/wiki/Bimodule

54 questions
6
votes
1 answer

Equivalent bimodule categories

Let $A,B$ be two rings such that their categories of bimodules are equivalent: $$A\mathsf{-Bimod} \simeq B\mathsf{-Bimod}$$ What can we say about $A$ and $B$? Are they isomorphic? Are they Morita-equivalent? Probably this has been studied in the…
Martin Brandenburg
  • 183,887
5
votes
0 answers

Example of Day convolution

By the Eilenberg-Watts theorem there is a monoidal equivalence between the category of $R^e$ -bimodules and the category of cocontinuous, additive endofunctors on the category of $R$ -bimodules. $R$ is a ring and $R^e=R\otimes \overline{R}$ its…
Margaret
  • 1,841
  • 5
  • 12
4
votes
0 answers

Normalized Hochschild complex

For a given field $k$ , let $A$ be an associative, unital $k$ -algebra. Let $M$ be an $A$ -bimodule. Define the Hochschild cochain complex of $A$ with coefficients in $M$ as $$C^n(A;M):=\operatorname{Hom}_k(A^{\otimes n},M)$$ with the usual…
Margaret
  • 1,841
  • 5
  • 12
4
votes
0 answers

Bimodules over a categorical monad

In category theory a monad consists in an endofunctor $M\colon \mathcal{C}\rightarrow \mathcal{C}$ together with natural transformations $\mu \colon M \circ M \Rightarrow M$ and $\eta \colon Id_{\mathcal{C}}\Rightarrow M$ satisfying the usual…
N.B.
  • 2,159
4
votes
1 answer

Does the category of $(R,S)$-bimodules contain a free object on one generator?

I believe that in the category of $(R,S)$ -bimodules where $R$ and $S$ are rings with identity, then ${}_R R \otimes_\mathbb{Z} S_S$ is a free object on one generator in this category. But, if $R, S$ are rings without identity, does the category of…
user02262019
  • 43
4
votes
1 answer

Is $Hom_k(A,N)$ a flat $A^e$-module?

For a field $k$, a $k$-algebra $A$ and an $A^e$-module $N$, where $A^e=A \otimes_k A^{op}$, I have proved that the $A^e$-module $Hom_k(A,N)$, with action $(a\otimes b) f (x) := a f(x) b$ is an injective $A^e$-module. Now I am trying to see if it is…
Marco Armenta
  • 952
4
votes
0 answers

Direct sum decomposition of bimodules

Let $R$ and $S$ be algebras over a commutative ring $k$. If $M$ is a left modules over $R$ and $N$ is a left module over $S$ then tensor product $M \otimes_k N$ is a bimodule i.e. left $R \otimes_k S$ module. Under which assumptions on $R$ and $S$…
Alex
  • 6,826
4
votes
2 answers

S,R bimodules subcategory of the category of S+R modules?

If $R$ and $S$ are commutative rings, then does the category $R \oplus S$-modules encompase the category of $(S,R)$-bimodules? I was thinking we can accomplish this by defining the action to be: $(r,s)\cdot x:=r\cdot (x\cdot s)$ and then doing…
AB_IM
  • 6,978
3
votes
1 answer

Bimodule over division rings simple on both sides

By a ring I mean an associative ring with multiplicative unity, and by a division ring I mean a skew field. My question is as follows Let $D, S$ be two division rings and $_DM_S$ be a $(D, S)$ -bimodule such that $_DM$ is simple left D-module and…
dmk
  • 585
  • 3
  • 13
3
votes
1 answer

Does $\mathrm{hom}_R(M,M)^{op}=\mathrm{hom}_{R^{op}}(M,M)$ if $M$ is a $(R,R)$-bimodule?

A $(R,S)$ -bimodule $M$ is the same as either a ring map $S^{op}\to \mathrm{hom}_R(M,M)$ or a ring map $R\to \mathrm{hom}_{S^{op}}(M,M)$ (i.e. $M$ can be viewed as $(S^{op},R^{op})$ -bimodule). Since any ring $R$ is an $(R,R)$ -bimodule, there are…
John
  • 127
3
votes
0 answers

2-category of bimodules in terms of spans?

There is an equivalence between modules over rings and Beck modules (Abelian group objects in a slice category) $$ \text{Mod}_R \cong \text{Ab}(\text{CRing}/R) $$ I think of this as similar to the relationship between presheafs and discrete…
user239001
3
votes
1 answer

Is a tensor product of finitely presented bimodules finitely presented?

Let $R$ be a commutative $k$ -algebra. In fact, I'm interested in the case when $k=\mathbb{Q}$ and $R$ is of the form $\mathbb{Q}[x_1,...,x_n]/I$ where $I$ is finitely generated. Recall that $R$ -bimodules are the same thing as $R \otimes_k R$…
Alvaro Martinez
  • 1,977
3
votes
0 answers

Tensor product $M_{n\times m}(K)\otimes_{M_{m\times m}(K)} M_{m\times n}(K)$

Let $M_{n\times m}(K)$ be the ring of $n\times m$ matrices over the field $K$ , $S=M_{n\times n}(K)$ , $R=M_{m\times m}(K)$ , $P=M_{n\times m}(K)$ $-$ (S,R)-bimodule, $Q=M_{m\times n}(K)$ $-$ $(R,S)$ -bimodule. I believe that $$P\otimes_{R} Q\cong…
SVG
  • 4,411
3
votes
0 answers

Characterizing fullness of a von Neumann algebra by the topology of its bimodules

Let $\mathcal{M}$ be a $\mathrm{II}_1$ factor. Among other characterizations, it is said to be full iff the adjoint map: $$ \mathrm{Ad}: U(\mathcal{M})/\mathbb{T} \longrightarrow \mathrm{Aut}(\mathcal{M}), $$ given by $\mathrm{Ad}([u])(x) = u x u^*$…
Adrián González Pérez
  • 2,436
3
votes
1 answer

Homomorphism of bimodules

Let $R,S,T$ be rings, $_RM_S$ be a $R$-$S$-bimodule and $_RN_T$ a $R$-$T$-bimodule. Besides we have $$H=\text{Hom}_R(M,N).$$ We can turn $H$ into a $S$-$T$-bimodule via $$(s.f)(m):=f(m.s)\quad\text{and}\quad (f.t)(m):=f(m).t \qquad \forall f\in…
user369147
1
2 3 4