nLab characteristic of a rig

Idea

The analogue of the characteristic of a ring, but for rigs.

Due to the possible failure of cancellativity for addition of a rig, it is insufficient to only consider the quotient rigs of $\mathbb{N}$ for the subrig generated by $1$, which is needed to characterize the additive $\mathbb{N}$-action on the rig for the definition of a characteristic.

Thus, unlike the characteristic of a ring, the characteristic of a rig is given by two natural numbers, one characterizing the subrigs of $\mathbb{N}$ and a second characterizing the quotient rigs of $\mathbb{N}$.

Terminology

In this article and in the literature on rig theory, this notion is just called a characteristic of a rig. However, to distinguish this notion of characteristic from the usual definition of a characteristic of a ring, perhaps the term rig characteristic can be used for this notion of characteristic for rings.

Definition

Every rig is an $\mathbb{N}$-module, so has a action $(n, x) \mapsto n x$ for natural number $n$ and element $x$.

Definition

A rig $R$ has characteristic $(m, n)$ for $m \geq 0$ and $n \geq 1$ if $m$ and $n$ are the smallest natural numbers such that for all elements $x$, $m x = (m + n) x$ . A rig $R$ has rig characteristic $(\infty,0)$ if there does not exist any positive natural number $n$ such that $m x = (m + n) x$.

There is another definition of a characteristic of a rig involving maps out of quotient rigs of $\mathbb{N}$.

Every quotient rig of $\mathbb{N}$ is equivalent to a quotient rig $\mathbb{N}/(x + m \sim x + m + n)$ for some natural number $m \geq 0$ and $n \geq 1$ (Rogers 2024). Thus, let $\mathbb{N}_{m, n}$ denote the quotient rig $\mathbb{N}/(x + m \simeq x + m + n)$.

Definition

A rig $R$ has characteristic $(m, n)$ for $m \geq 0$ and $n \geq 1$ if the subrig of $R$ generated by $1$ is isomorphic to $\mathbb{N}_{m, n}$, and a rig $R$ has characteristic $(\infty,0)$ if the subrig of $R$ generated by $1$ is isomorphic to $\mathbb{N}$.

Properties

Every ring with positive characteristic $n$ has, as a rig, a rig characteristic of $(0, n)$. Moreover, every rig with rig characteristic $(0, n)$ is in fact a ring with positive characteristic $n$.

Similarly, every ring with characteristic zero has rig characteristic $(\infty,0)$ as a rig. However, there still exist rigs which are not rings with characteristic zero, such as the natural numbers $\mathbb{N}$.

More generally, every rig whose addition is a cancellative monoid has rig characteristic $(0, n)$ or rig characteristic $(\infty,0)$.

Related concepts

• characteristic

• characteristic zero

• positive characteristic

References

• Morgan Rogers, From free idempotent monoids to free multiplicatively idempotent rigs [arXiv:2408.17440]