nLab possibly trivial integral domain

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

If RR is the trivial ring, DD is the empty subset, the ideal generated by the empty subset is the trivial ideal 0R0R, and R/0RR/0R is always isomorphic to RR.

 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 RR be a commutative ring, let DRD \subseteq R be the subset of all zero divisors or non-cancellative elements in DD, and let I DI_D be the two-sided ideal generated by DD. The quotient of RR by I DI_D is the possibly trivial integral domain of RR.

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

References

The concept of a possibly trivial integral domain appeared in

where the possibly trivial discrete integral domains are simply called “integral domains”.

Last revised on December 9, 2022 at 03:10:16. See the history of this page for a list of all contributions to it.