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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)
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)
How is this article related to Bilinear map? It seems there is an overlap. Nageh ( talk) 16:22, 26 May 2010 (UTC)
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)
For a pairing to be perfect, I think, one should require both induced maps, i.e. also , to be isomorphisms. This does not follow automatically in general. Take for example the pairing which (as far as I know) should not be considered a perfect pairing, however, the -dual of is trivial, so is an isomorphism. 89.13.163.119 ( talk) 20:32, 30 December 2014 (UTC)
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)
@ 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)
This article is rated Start-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | ||||||||||||||
|
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)
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)
How is this article related to Bilinear map? It seems there is an overlap. Nageh ( talk) 16:22, 26 May 2010 (UTC)
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)
For a pairing to be perfect, I think, one should require both induced maps, i.e. also , to be isomorphisms. This does not follow automatically in general. Take for example the pairing which (as far as I know) should not be considered a perfect pairing, however, the -dual of is trivial, so is an isomorphism. 89.13.163.119 ( talk) 20:32, 30 December 2014 (UTC)
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)
@ 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)