Talk:Pairing

note1

Pairings used cryptography allow the output to be R-module too. I will amend the article, unless there are objections. DRLB (talk) 23:27, 23 January 2008 (UTC)

note2

I Remove the Italian reference because is pointing at the Bluetooth pairing instead of the mathematical concept.--Liberac (talk) 17:46, 6 August 2009 (UTC)

Bilinear map

How is this article related to Bilinear map? It seems there is an overlap. Nageh (talk) 16:22, 26 May 2010 (UTC)

I suspect that pairing may be a direct synonym for bilinear form, perhaps an alternate term used in disciplines such as algebraic geometry. As such, it should be merged there, with this becoming a redirect. —Quondum 03:36, 18 October 2016 (UTC)

A real intro please?

I have absolutely no idea what this is about. I suspect, from the name, that it has to do with pairing up mathematical entities, but none of the editors seems inclined to actually describe this in a simple fashion. Could someone attack this please? Maury Markowitz (talk) 10:52, 19 September 2012 (UTC)

Perfect pairing

For a pairing to be perfect, I think, one should require both induced maps, i.e. also $N\to \operatorname {Hom} _{R}(M,L)$, to be isomorphisms. This does not follow automatically in general. Take for example the pairing $\{0\}\times \mathbb {Q} \to \mathbb {Z}$ which (as far as I know) should not be considered a perfect pairing, however, the $\mathbb {Z}$-dual of $\mathbb {Q}$ is trivial, so $\{0\}\to \operatorname {Hom} _{\mathbb {Z} }(\mathbb {Q} ,\mathbb {Z} )$ is an isomorphism.89.13.163.119 (talk) 20:32, 30 December 2014 (UTC)

General modules

The concept of pairing is not specific to commutative rings; it applies for any modules (see e.g. Bourbaki). The restriction of the base ring to being commutative should be removed from the definition. This does mean that left and right modules must be called out and the premise of bilinearity must be restated, but this is straightforward and easily referenced. —Quondum 15:45, 6 October 2016 (UTC)

Bilinearity

@Quondum: A good point about inner product not necessarily bilinear. I think the idea here is that a complex inner product is a real inner product (which is an R-bilinear map) satisfying a certain property about multiplication by complex scalars. (It is not rare that a pairing satisfies some additional property). Of course, this need to be clarified in the article. —- Taku (talk) 04:52, 7 February 2021 (UTC)

Unfortunately, "inner product" often gets used in WP in the conflicting sense of a bilinear form (not necessarily positive definite), especially in geometric algebra and physics article. Given that the use of inner product here would need to be caveated (e.g. restricted to real vector spaces) and often is ambiguous elsewhere in WP, it would probably be better to remove the mention of inner product. Incidentally, the definition conflicts with the lead: is it a map to the underlying ring or to a module over the underlying ring? —Quondum 16:38, 7 February 2021 (UTC)

I took the leap and updated the evidently outdated lead to reflect the body, but suspect that it is still not an adequate general definition. EoM defines a pairing as "a mapping defined on the Cartesian product of two sets", and says that additional requirements such as bilinearity may be imposed depending on context. I have not found other definitions outside the context of cryptography, which often imposes Z-bilinearity. —Quondum 19:52, 9 February 2021 (UTC)