nLab possibly trivial integral domain

Idea

While in general, integral domains are non-trivial rings, it is sometimes useful in commutative algebra for the trivial ring to be an integral domain. Thus, the idea of a possibly trivial integral domain.

 Definition

Let $R$ be a commutative ring and let $D \subseteq R$ be the subset of all zero divisors or non-cancellative elements in $D$. A possibly trivial integral domain is a commutative ring $R$ such that every element in the two-sided ideal $I_D$ generated by $D$ is equal to zero, or that the quotient ring $R/I_D$ is isomorphic to $R$.

If $R$ is the trivial ring, $D$ 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 integral domain from a commutative ring

From every commutative ring, one could construct a possibly trivial integral domain.

As above, let $R$ be a commutative ring, let $D \subseteq R$ be the subset of all zero divisors or non-cancellative elements in $D$, and let $I_D$ be the two-sided ideal generated by $D$. The quotient of $R$ by $I_D$ is the possibly trivial integral domain of $R$.

$R/I_D$ is a possibly trivial integral domain, because the quotient of $R$ by $I_D$ takes all elements of $I_D$ to zero. If $R$ is an approximate integral domain, then $R/I_D$ is an integral domain; otherwise, $R/I_D$ is the trivial ring.

In constructive mathematics

In constructive mathematics, the notion of integral domain bifurcates into multiple notions. While the definition given above results in the usual notion of an integral domain as a commutative ring whose zero divisors are equal to zero, there are also possibly trivial Heyting integral domains and possibly trivial discrete integral domains, which refer directly to cancellative elements.

Definition

A possibly trivial integral domain $R$ is Heyting if it has a tight apartness relation and where every element apart from zero is cancellative. The trivial ring is trivially Heyting because the set of elements apart from zero is the empty subset.

Definition

A possibly trivial integral domain is discrete if every element of $R$ is either equal to zero or cancellative. The trivial ring is trivially discrete because every element is both equal to zero and cancellative. These are simply called integral domains in LombardiQuitté2010. Possibly trivial discrete integral domains have decidable equality.

 See also

• commutative ring
• integral domain
• possibly trivial field

References

The concept of a possibly trivial integral domain 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 discrete integral domains are simply called “integral domains”.