A pairing is any R-bilinear map . That is, it satisfies
- ,
- and
for any and any and any . Or equivalently, a pairing is an R-linear map
where denotes the tensor product of M and N.
A pairing can also be considered as an R-linear map, which matches the first definition by setting .
A pairing is called perfect if the above map is an isomorphism of R-modules.
If a pairing is called alternating if for the above map we have .
A pairing is called non-degenerate if for the above map we have that for all implies .
Read more about Pairing: Examples, Pairings in Cryptography, Slightly Different Usages of The Notion of Pairing
Famous quotes containing the word pairing:
“Through man, and woman, and sea, and star,
Saw the dance of nature forward far;
Through worlds, and races, and terms, and times,
Saw musical order, and pairing rhymes.”
—Ralph Waldo Emerson (1803–1882)