nLab Pythagorean ring




Pythagorean rings are rings in which all elements satisfy the common Pythagorean identity, generalizing from the real numbers.


Let RR be a (possibly nonassociative and/or possibly nonunital) ring. Then RR is a Pythagorean ring if it has a binary operation p:R×RRp:R \times R \to R such that for every element aa and bb in RR, a 2+b 2=p(a,b) 2a^2 + b^2 = p(a,b)^2. (Note all binary operations \cdot have an associated cartesian square () 2(-)^2 defined as a 2=aaa^2 = a \cdot a.)

There is also a nn-ary version of pp, which is a finite sum

i=0 na i 2=p n(a 0,a 1,,a n) 2\sum_{i=0}^n a_i^2 = p_n(a_0,a_1,\ldots,a_n)^2

for a natural number n:n:\mathbb{N}.


Due to the nature of addition in an abelian group, the set RR with the binary operation pp is a commutative semigroup.

Every finitely generated RR-module AA for a Pythagorean ring RR by a set of finite cardinality nn has an absolute value |()|:AR\vert (-) \vert: A \to R given by the nn-ary version of pp in RR for the scalars a ia_i of an element aa in AA, and a quadratic form given by |()| 2\vert(-)\vert^2. As a result, for every finitely generated RR-module AA for a Pythagorean ring RR there is an associated Clifford algebra Cl(R,|()| 2)Cl(R,\vert(-)\vert^2).


  • The real numbers are a Pythagorean ring.

  • A Pythagorean ring that is a field is a Pythagorean field.

Last revised on June 5, 2021 at 10:41:27. See the history of this page for a list of all contributions to it.