Pairing

The concept of pairing treated here occurs in mathematics.

Definition

Let R be a commutative ring with unity, and let M, N and L be three R-modules.

A pairing is any R-bilinear map . That is, it satisfies

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 .

Examples

Any scalar product on a real vector space V is a pairing (set M = N = V, R = R in the above definitions).

The determinant map (2 × 2 matrices over k) → k can be seen as a pairing .

The Hopf map written as is an example of a pairing. In for instance, Hardie et al. present an explicit construction of the map using poset models.

Pairings in cryptography

Pairing (computing)

Pairing, sometimes known as bonding, is a process used in computer networking that helps set up an initial linkage between computing devices to allow communications between them. The most common example is used in Bluetooth, where the pairing process is used to link devices like a bluetooth headset with a mobile phone.