nLab integral domain

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Context

Algebra

higher algebra

universal algebra

Contents

Definition

Some authors do not require an integral domain to be commutative. However, on the nLab we require our integral domains to be commutative.

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 commutative ring is non-trivial if $0 \neq 1$.

Definition

An integral domain is a non-trivial commutative ring $R$ such that 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 $R$.

Since every element of the ideal generated by the set of zero divisors is equal to zero, and the integral domain is a non-trivial ring, which means that there is at least one zero divisor, it follows that zero is a zero divisor and if $a \cdot b = 0$, then $a = 0$ or $b = 0$. It then follows that every integral domain is a reduced ring.

The ring of fractions of every integral domain is a field, the field of fractions.

The ring of formal power series of an integral domain is a local integral domain.

In constructive mathematics

in constructive mathematics, there are different inequivalent ways to define an integral domain. The above definition is sometimes called a ring without zero divisors. Stronger notions include:

Definition

A Heyting integral domain is an integral domain which has a tight apartness relation and where every element apart from zero is regular. The ring of fractions of a Heyting integral domain is a Heyting field.

Definition

A discrete integral domain is a Heyting integral domain with decidable equality, or a Heyting integral domain where every element is either zero or regular. The ring of fractions of a discrete integral domain is a discrete field.

Rings without zero divisors and discrete integral domains are both definable in coherent logic. However, Heyting integral domains can only be defined in first-order logic.

Generalizations

Sometimes, one may want the trivial ring to be an integral domain, resulting in the notion of possibly trivial integral domain. In LombardiQuitté2010, the authors define an integral domain to be a possibly trivial integral domain.

In principle, one could just as easily consider a commutative rig or commutative semiring $R$. In that case, however, only the definition involving the cancellative property extends to rigs and semirings. Furthermore, we should add the additional requirement that addition in $R$ is cancellable (that is, addition by any element is injective), to make the analogue of the previous paragraph correct. Since these could also be done for general domains to turn them into what could be called domain rigs and domain semirings, these could be called commutative domain rigs or commutative domain semirings.