nLab Pythagorean ring

Context

Algebra

higher algebra

universal algebra

Contenta

Idea

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

Definitions

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

There is also a $n$-ary version of $p$, which is a finite sum

$\sum_{i=0}^n a_i^2 = p_n(a_0,a_1,\ldots,a_n)^2$

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

Properties

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

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

Examples

• The real numbers are a Pythagorean ring.

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