Warfield Domains: Module Theory from Linear Algebra to Commutative Algebra Through Abelian Groups

Abstract

In this survey paper we illustrate the generalizations to modules over integral domains of the basic result of linear algebra stating that every finite dimensional vector space is canonically isomorphic to its bidual. These generalizations culminate in the discovery of Warfield domains. We present the characterization of Warfield domains in terms of properties of their overrings, as well as their intrinsic characterizations in the classical cases of Noetherian and integrally closed domains, and in the general case. Finally, the development of the results on direct sum decompositions of torsionless modules from the classical case of Dedekind domains up to Warfield domains is sketched; this subject was originated by the 1912 result of Steinitz on finite direct sums of ideals in algebraic number rings.

Abelian varieties isogenous to a power of an elliptic curve

“…See [BF60]. For generalizations to other integral domains, see [Bas63, §7], [BF65], [Lev85], and the survey article [Sal02]. ◻…”

On Overrings of Gorenstein Dedekind Domains

“…The following theorem is [19,Theorem 3.1]. From this, it can be seen that a Noetherian domain is divisorial if and only if it is reflexive.…”

“…On the other hand, if any overring of a G-Dedekind domain R is still a G-Dedekind domain, then R must be totally divisorial and further, totally reflexive since it is a Noetherian domain. Therefore, by the above two theorems or [3,Theorem 7.3], R must be a Warfield domain and all the ideals of R are 2-generated (the definition of a Warfield domain can be seen in [19]). Next, we will prove that the G-Dedekind domain Q + X 2 Q[X] is a Warfield domain.…”

“…In Section 2, we give an example which shows that this is not necessarily the case. In fact, it can be seen from [19,Theorem 6.3] and [19,Theorem 6.4] that a Noetherian domain has the property that each of its overring is a G-Dedekind domain if and only if it is a Warfield domain. In [5] and [6], Driss Bennis and Najib Mahdou introduced strongly Gorenstein projective modules and n-strongly Gorenstein projective modules respectively.…”

“…The author in [19] gave a general introduction to Warfield domains. A domain R is called a Warfield domain if, given any submodule A of the field of quotients K, the A-torsionless End R (A)-modules of finite rank are A-reflexive.…”