nLab approximate integral domain

Idea

A weaker notion of integral domain which allows for some zero divisors, but for which one may quotient out the zero divisors to obtain an integral domain.

“Approximate integral domain” is a placeholder name for a concept which may or may not have another name in the mathematics literature. The idea however is that approximate integral domains are to integral domains as local rings are to Heyting fields, and as weak local rings are to weak Heyting fields.

 Definition

An approximate integral domain is a commutative ring $R$ such that:

• $R$ is nontrivial ($0 \ne 1$); and

• The zero divisors form an ideal. (equivalently, the non-cancellative elements form an ideal).

Thus, the quotient of an approximate integral domain by its ideal of zero divisors is an integral domain.

Every approximate integral domain has an equivalence relation $\approx$, defined as $x \approx y$ if and only if $x - y$ is a zero divisor. Hence the name “approximate” integral domain. Then integral domains are precisely the approximate integral domains for which $\approx$ implies equality.

In constructive mathematics

In constructive mathematics, similar to the notion of local ring, integral domain, and field, the notion of approximate integral domain bifurcates into multiple distinct notions:

A weak approximate integral domain is an approximate integral domain defined as above.

Recall that a cancellative element in a commutative ring $R$ is an element $a \in R$ for which both left and right multiplication by $a$ is an injection, and zero divisors are precisely the elements in $R$ which are not cancellative.

A strict approximate integral domain is a weak approximate integral domain for which additionally the cancellative elements form an anti-ideal:

• $0$ is not cancellative;
• if $a + b$ is cancellative, then either $a$ is cancellative or $b$ is cancellative;
• if $a \cdot b$ is cancellative, then $a$ is cancellative and $b$ is cancellative (this is trivially true in any commutative ring).

The quotient ring of a strict approximate integral domain by its anti-ideal of cancellative elements is a Heyting integral domain.

One can define an apartness relation in any strict approximate integral domain: $x \# y$ iff $x - y$ is cancellative. Then the approximate integral domain is a Heyting integral domain if and only if this apartness relation is tight.

Proposition

The addition and multiplication operations on a strict approximate integral domain $R$ are strongly extensional with respect to the canonical apartness relation $\#$ defined by $x \# y$ iff $x - y$ is cancellative. In this way a strict approximate integral domain becomes an internal ring object in the category $Apart$, consisting of sets with apartness relations and maps (strongly extensional functions) between them.

Proof

Recall that products $X \times Y$ in the category of sets with apartness relations is the cartesian product of the underlying sets equipped with the apartness relation defined by $(x, y) \# (x', y')$ iff $x \# x'$ in $X$ or $y \# y'$ in $Y$. Recall also that a function $f: X \to Y$ between sets with apartness relations is strongly extensional if $f(x) \# f(y)$ implies $x \# y$.

For addition, if $(x + y) \# (x' + y')$, then $x + y - (x' + y') = (x - x') + (y - y')$ is cancellative, so $x - x'$ or $y - y'$ is cancellative since $R$ is a strict approximate integral domain, whence $(x, y) # (x', y')$. Thus addition is strongly extensional.

For multiplication, if $x y # x' y'$, then $x y - x' y'$ is cancellative. Write $x y - x' y' = (x - x')y + x'(y - y')$. Since $R$ is a strict approximate integral domain, either $(x - x')y$ is a cancellative element or $x'(y - y')$ is a cancellative element. From this we easily conclude $x - x'$ is a cancellative element or $y - y'$ is, since cancellative elements are closed under multiplicaiton, whence $(x, y) # (x', y')$. So multiplication is also strongly extensional.

Theorem

For a strict approximate integral domain, the ring of fractions obtained by inverting the cancellative elements is a local ring.

 Examples and non-examples

• The integers are an approximate integral domain which are an integral domain.

• The dual integers $\mathbb{Z}[\epsilon]/\epsilon^2$ is an approximate integral domain where the nilpotent infinitesimal $\epsilon \in \mathbb{Z}[\epsilon]/\epsilon^2$ is a non-zero zero divisor.

• For any prime number $p$ and any positive natural number $n$, the prime power local ring $\mathbb{Z}/p^n\mathbb{Z}$ is an approximate integral domain, whose ideal of zero divisors is the ideal $p(\mathbb{Z}/p^n\mathbb{Z})$. The quotient of $\mathbb{Z}/p^n\mathbb{Z}$ by its ideal of zero divisors is the finite field $\mathbb{Z}/p\mathbb{Z}$, indicating that it is also a weak local ring.

• There exist commutative rings which are not approximate integral domains. For example, the integers modulo 6 $\mathbb{Z}/6\mathbb{Z}$ is not an approximate integral domain, because $3$ and $4$ are both zero divisors, but $3 + 4$ is cancellative. When one tries to quotient out the zero divisors, the resulting ring is trivial.

 See also

• integral domain

• ordered integral domain

• weak local ring

• local ring