nLab reciprocal algebra

Idea

An algebra with a reciprocal that behaves like 1x\frac{1}{x} in the rational and real numbers.

Definition

A reciprocal algebra is a possibly non-associative algebra AA, typically over a field kk, which has the property that for non-zero element bb, there exists an element cc, called the reciprocal of bb, such that for every element aa

  • a(cb)=aa (c b) = a
  • (cb)a=a(c b) a = a
  • a(bc)=aa (b c) = a
  • (bc)a=a(b c) a = a

Examples

Counter-examples

  • There exists a division algebra with identity that does not have two-sided inverses for every nonzero element.

Last revised on June 13, 2022 at 15:38:55. See the history of this page for a list of all contributions to it.