nLab Pythagorean ring

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.

Related concepts

• Euclidean geometry

• absolute value

• Clifford algebra

• exponential ring

External links

• Wikipedia, Pythagorean field