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Warfield domains: module theory from linear algebra to commutative algebra through abelian groups. (English) Zbl 1054.13005

In this well-written survey, the author has three main goals. The first goal is to outline the chronological development of the theory of reflexive and related domains from the sixties to the present days. The second goal is to illustrate the various characterizations of Warfield domains: the non-intrinsic ones through the reflexivity of their overrings, and the intrinsic ones in the classical cases of Noetherian and integrally closed domains, and in the general case. The third goal is to present a series of results on the good behaviour of the torsionless modules of finite rank over Warfield domains with respect to direct sum decompositions. This subject goes back to a pioneering theorem in Steinitz’s 1912 paper [cf. E. Steinitz, Math. Ann. 72, 297–345 (1912; JFM 43.0274.01)].

MSC:

13C05 Structure, classification theorems for modules and ideals in commutative rings
13-03 History of commutative algebra
01A60 History of mathematics in the 20th century
01A61 History of mathematics in the 21st century

Citations:

JFM 43.0274.01
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