# Zulip Chat Archive

## Stream: maths

### Topic: bilinear forms

#### Yury G. Kudryashov (Jun 01 2020 at 23:04):

Is there a definition that generalizes both bilinear forms `M₁ → M₂ → R`

where `M₁`

and `M₂`

are modules over a ring `R`

and bilinear forms `M₁ → M₂ → M₃`

where `M₁`

, `M₂`

, and `M₃`

are modules over a **commutative** ring `R`

?

#### Oliver Nash (Jun 02 2020 at 07:42):

I don't have an answer to your question Yury. I just wanted to say I'm very glad to see that someone is thinking about such generalisations. I wasn't aware of this thread and spent a few minutes a few days ago messing around with something like:

```
structure bilinear_pairing :=
(bilin : M₁ → M₂ → R)
(...)
variables (I : ring_anti_equiv R R)
def anti_module : module R M₁ :=
{ smul := λ r x, (I r) • x,
...}
def sesq_form' := @bilinear_pairing R M₁ M₁ _ _ _ _ (anti_module R M₁ I)
```

#### Reid Barton (Jun 02 2020 at 13:50):

In the former case shouldn't `M₁`

be a right `R`

-module?

#### Yury G. Kudryashov (Jun 02 2020 at 18:02):

A right `R`

-module is a left `opposite R`

-module. The question is what to do with the codomain. Should it carry be a bi-module (`R`

-module + `opposite R`

-module + actions commute)?

#### Yury G. Kudryashov (Jun 02 2020 at 18:04):

@Oliver Nash Note that #2884 drops `ring_anti_equiv`

in favour of `R ≃+* Rᵒᵖ`

.

Last updated: May 12 2021 at 08:14 UTC