nLab characteristic

Idea

It is well known that you cannot divide by zero, lest you be doomed to triviality. Conversely, in a field, you can divide by anything except zero. But this rule can be misleading, since it's possible that (even) an ordinary number can be zero when you don't expect it! The characteristic of a field states when (if ever) this happens.

It is straightforward to generalise from fields to other rings. See rig characteristic for the corresponding notion of a characteristic of a rig.

See also characteristic zero.

Definition

For rings

Every ring is an $\mathbb{Z}$-module, so has a action $(n, x) \mapsto n x$ for integer $n$ and element $x$.

Definition

A ring $R$ has characteristic $n$ for positive integer $n \geq 1$ if $n$ is the smallest positive integer such that for all elements $x$, $n x = 0$. A ring $R$ has characteristic $0$ if there does not exist any positive integer $n$ such that $n x = 0$.

There is another definition involving the kernel of the unique ring homomorphism from $\mathbb{Z}$:

Let $K$ be a ring (possibly a commutative ring, possibly even a field). Then there exists a unique ring homomorphism $\phi_K\colon \mathbb{Z} \to K$ to $K$ from the initial ring, which is the ring $\mathbb{Z}$ of integers. The kernel of $\phi_K$ is an ideal of $\mathbb{Z}$, which (by a well-known property of $\mathbb{Z}$) is a principal ideal with a non-negative generator and a non-positive generator. The non-negative generator is the characteristic of $K$, denoted $\char K$.

For $E_\infty$-rings

The concept of the characteristic has been generalized to E-∞ rings (Szymik 12, Szymik 13, Baker 14).

Properties

If $n$ is a natural number, then we suppress mention of $\phi_K$ to think of $n$ as an element of $K$. If $K$ is a ring, then we do the same for a negative integer $n$. We then have that $n = 0$ in $K$ if and only if $n$ is a multiple of $\char K$.

The characteristic of a field must be either zero or a prime number. Basically, this is because the kernel of $\phi_K$, for $K$ a field, must be a prime ideal. Similarly, the characteristic of an integral domain must be either zero or a prime number.

If there is any homomorphism at all between two fields, then they have the same characterstic. In other words, any extension of a field keeps the same characteristic.

Relation with the rig characteristic

Every ring with positive characteristic $n$ has 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 zero. However, there still exist rigs which are not rings with rig characteristic zero, such as the natural numbers $\mathbb{N}$.

Examples

If $n$ is a positive natural number, then the characteristic of $\mathbb{N}/n = \mathbb{Z}/n$ is $n$. This ring is always a commutative ring, and it is a field if and only if $n$ is prime, in which case it is the prime field $\mathbb{F}_n$. More generally, every finite field has positive prime characteristic.

For $n = 0$, $\mathbb{N}/0 = \mathbb{N}$, $\mathbb{Z}/0 = \mathbb{Z}$, and the prime field $\mathbb{F}_0 = \mathbb{Q}$ (the field of rational numbers) are no longer all the same, but they still all have characteristic $0$. Every ordered field has characteristic $0$. The real numbers and complex numbers each form fields of characteristic $0$.

Related concepts

• freshman's dream

• characteristic zero, positive characteristic

• rig characteristic

References

• Wikipedia, Characteristic (algebra)

Discussion for E-∞ rings:

• Markus Szymik, Commutative S-algebras of prime characteristics and applications to unoriented bordism (arXiv:1211.3239)

• Markus Szymik, String bordism and chromatic characteristics (arXiv:1312.4658)

• Andrew Baker, Characteristics for $E_\infty$ ring spectra (arXiv:1405.3695)