nLab possibly trivial field

Idea

While in general, fields are non-trivial rings, it is sometimes useful in commutative algebra for the trivial ring to be a regarded as a field. Thus, the idea of a possibly trivial field.

Definition

Let $R$ be a commutative ring, let $\mathrm{Reg}(R)$ be the monoid of regular elements in $R$, let $\neg \mathrm{Reg}(R)$ be the complement of $\mathrm{Reg}(R)$, the set of zero divisors in $R$, and let $I_{\neg \mathrm{Reg}(R)}$ be the two-sided ideal generated by $\neg \mathrm{Reg}(R)$.

A possibly trivial field is a commutative ring $R$ such that

• every element in $\mathrm{Reg}(R)$ is a unit, or that the localization $R[\mathrm{Reg}(R)^{-1}]$ is isomorphic to $R$.

• every element in $I_{\neg \mathrm{Reg}(R)}$ is equal to zero, or that the quotient ring $R/I_{\neg \mathrm{Reg}(R)}$ is isomorphic to $R$.

If $R$ is the trivial ring, $\neg \mathrm{Reg}(R)$ is the empty subset, the ideal generated by the empty subset is the trivial ideal $0R$, and $R/0R$ is always isomorphic to $R$.

Constructing a possibly trivial field from a commutative ring

Let $R$ be a commutative ring, let $\mathrm{Reg}(R)$ be the monoid of regular elements in $R$, let $\neg \mathrm{Reg}(R)$ be the complement of $\mathrm{Reg}(R)$, the set of zero divisors in $R$, and let $I_{\neg \mathrm{Reg}(R)}$ be the two-sided ideal generated by $\neg \mathrm{Reg}(R)$.

There are two operations one could do to a commutative ring $R$, taking the quotient by $I_{\neg \mathrm{Reg}(R)}$, and localizing at $\mathrm{Reg}(R)$. Taking the quotient by $I_{\neg \mathrm{Reg}(R)}$ results in a possibly trivial integral domain $R/I_{\neg \mathrm{Reg}(R)}$, while localizing at $\mathrm{Reg}(R)$ results in a prefield ring, the ring of fractions $R[\mathrm{Reg}(R)^{-1}]$. The operations are commutative: first taking the quotient by $I_{\neg \mathrm{Reg}(R)}$ then localizing at $\mathrm{Reg}(R)$ and first localizing at $\mathrm{Reg}(R)$ then taking the quotient by $I_{\neg \mathrm{Reg}(R)}$ results in isomorphic possibly trivial fields, because no element is both a zero divisor and regular.

Thus, one has a functor $L:\mathrm{CRing} \to \mathrm{Prefield}$ which takes a commutative ring to its ring of fractions, and a functor $Q:\mathrm{CRing} \to \mathrm{PTIntDom}$ taking a commutative ring to its quotient possibly trivial integral domain, and because the category of prefield rings and the category of possibly trivial integral domains are subcategories of the category of commutative rings, there are also subfunctors $L':\mathrm{PTIntDom} \to \mathrm{PTField}$ and $Q':\mathrm{Prefield} \to \mathrm{PTField}$ where $L' \subseteq L$ and $Q' \subseteq Q$. The functors form a commutative square with natural isomorphism $p:L' \circ Q \cong Q' \circ L$:

$$\array{& \mathrm{CRing} & \overset{L}\rightarrow & \mathrm{Prefield} & \\ Q & \downarrow & \cong &\downarrow & Q'\\ & \mathrm{PTIntDom} & \underset{L'}\rightarrow& \mathrm{PTField} & \\ }$$

In constructive mathematics

In constructive mathematics, the notion of field bifurcates into multiple notions. The definition given above results in a possibly trivial weak Heyting field. However, there are also possibly trivial Heyting fields and possibly trivial discrete fields.

Possibly trivial Heyting fields

A possibly trivial field $R$ is Heyting if, whenever the sum of two elements $a + b$ is invertible, then either $a$ is invertible or $b$ is invertible. These are simply called Heyting fields in LombardiQuitté2010.

Possibly trivial discrete fields

A possibly trivial field is discrete if every element is either zero or invertible. These are simply called discrete fields in LombardiQuitté2010. Possibly trivial discrete fields have decidable equality.

See also

• commutative ring
• field
• possibly trivial integral domain
• weak local ring

References

The concept of a possibly trivial field appeared in

• Henri Lombardi, Claude Quitté (2010): Commutative algebra: Constructive methods (Finite projective modules) Translated by Tania K. Roblo, Springer (2015) (doi:10.1007/978-94-017-9944-7, pdf)

where the possibly trivial Heyting fields are called “Heyting fields” and the possibly trivial discrete field are called “discrete fields”.