Galois Theory and Module Theory

Galois Theory and Module Theory

            Let K is a field, G a finite group of automorphisms, and let F is be corresponding Galois subfield.  A normal basis for K over the field F is a vector space basis consisting of the conjugates under G of an element u of K. Then u is said to generate a normal basis. This is equivalent to saying that K, as a natural module over the group ring FG, is isomorphic to FG as a right FG-module. If f is the FG-isomorphism of FG with K, then u = f(1) generates a normal basis.  That one was able to use the structure of the group FG in field theory struck me profoundly when I was working on my Ph. D. thesis at Purdue under the direction of Sam Perlis.

Proofs of the Existence of a Normal Basis

            Many proofs of the existence of a normal basis exist. M. Deuring [1932] and R. Stauffer [1936] proved it when the characteristic of F does not divide the order of G, which is equal to the dimension of K over F, E. Artin [1948] gave a simple proof when F Is infinite, J. W. S. Cassells & G. E. Wall [1950] proved it when F is finite, and Nakayama [1940] proved it for skew fields. Also in [1941] Nakayama extends the results of Deuring and Stauffer to include their exceptional case.

Completely Basic Elements

            Refining the methods of Artin, in my 1955 Purdue U. Ph. D. thesis I proved the existence of a completely basic element, that is, an element u such that u generates a normal basis for K over every intermediated field between K and F. This, and the references above, appeared in Extensions of normal bases and completely basic fields in the 1957 Trans. of the Amer. Math. Soc.

                        Group rings led me to consider not only module theory but also quasi-Frobenius (QF) rings, since any group ring FG of a finite group over any field F is QF, in fact, Frobenius. A QF ring is characterized by the property that every injective right R-module is projective (theorem of Faith-Walker, see Rings & Things, Th. 3.5 B), and also by the dual property that every projective right R-module is injective (of theorem of mine, “R &T,” Th. 3.5 C.

            The proof of the Faith-Walker theorem led to the fruitful module- theoretical concept of ∑-injective modules, and to our:  Theorem. A ring R is right Noetherian iff every injective right R-module is ∑-injective (see, e.g., “R & T,” Theorem 3.5 A, or see “Boyle’s Theorem, Conjecture, and ∑-injective Modules,” Document # 10 above.)  This is used in the proof of the previous Th. 3.5 B. 

Pseudo-Frobenius Rings

            Galois Theory led not only to QF rings, but also to their successors, right Pseudo-Frobenius (PF) rings. These were first introduced by Goro Azumaya who called them “upper distinguished rings.” See, for example, vol. 2 of my Springer-Verlag Algebra. Osofsky, however, give the first example of a right but not left PF ring. Results of Azumaya, Osofsky and Utumi on PF rings also are summarized in “R & T,” Theorem 4.20. Moreover, my recent paper, Factor Rings of Pseudo-Frobenius Rings, J. Algebra and Its Applications, 5(2006) 847-854 (Erratum, ibid. 6(2007), is on this subject.

Finitely Pseudo Frobenius Rings

            A characterizing property of right PF ring is that every faithful right R-module generates the category mod-R of all right R-modules (see, e.g, “R & T,” Theorem 4.20.)  A ring R is right finitely pseudo-Frobenius (right FPF) if every finitely generated faithful right R-module generates the category mod-R of all right R-modules.. 

            S. Endo originated the FPF concept back in 1967.  Moreover,  characterized all commutative FPF rings in 1982:  Theorem. A commutative ring R is FPF iff : (1) every finitely generated faithful ideal  of R is projective; and (2) R has self-injective classical quotient ring Q(R)-- see, “R & T,” Theorem 5.42. The results for non-commutative right FPF rings are summarized in “R & T,” Theorems 4.26-4.32.