symmetric monoidal (∞,1)-category of spectra
Pythagorean rings are rings in which all elements satisfy the common Pythagorean identity, generalizing from the real numbers.
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
for a natural number $n:\mathbb{N}$.
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 $Cl(R,\vert(-)\vert^2)$.
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.