The regular element property

The property that an ideal whose annihilator is zero contains a regular element is examined from the point of view of constructive mathematics. It is shown that this property holds for nitely presented algebras over discrete elds, and for coherent, Noetherian, strongly discrete rings that contain an in nite eld. Let R be a commutative ring and M an R-module. For any subset I of R, we write AM(I) = fx 2M : Ix = 0g for the annihilator of I in M , which is a submodule of M . An element r in R is said to be M-regular if AM(r) = 0. We say that R has the regular element property (REP) if for each nitely presented R-module M , and nitely generated ideal I, if AM(I) = 0, then I contains an M -regular element. According to Kaplansky, [2, page 65] the theorem that a Noetherian ring has the REP is \a result that is among the most useful in the theory of commutative rings." In particular, it is basic to the characterization of depth, for local rings with residue class eld k, as the least n such that Ext(k;M) 6= 0: it says that if Ext(k;M) = 0, then the depth of M is at least 1. We will investigate this theorem from the point of view of constructive mathematics in the sense of Bishop [1], that is, mathematics done in the context of intuitionistic logic. The casual reader should be able to follow the discussion by thinking in terms of computations and constructions. 1. A constructive look at Noetherian rings First, let's look at the main ideas involved from this point of view. A ring R is Noetherian if, given a chain of nitely generated ideals