Galois theory for general rings with minimum condition

TADASI NAKAYAMA

doi:10.2969/jmsj/00130203

TADASI NAKAYAMA

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JOURNAL FREE ACCESS

1949 Volume 1 Issue 3 Pages 203-216

DOI https://doi.org/10.2969/jmsj/00130203

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Published: 1949 Received: February 11, 1948 Available on J-STAGE: August 29, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: August 29, 2006 Reason for correction: - Correction: CITATION Details: Right : 1) N. Jacobson, The fundamental theorem of Galois theory for quasi-fields, Ann. Math. 41 (1940).
2) G. Azumaya, New foundation for the theory of simple rings, forthcoming in Proc. Imp. Acad. Japan: G. Azumaya, Galois theory for uni-serial rings, Journ. Math. Soc. Japan 1 (1949).
3) Another extreme case is that of (closed) irreducible rings. See T. Nakayama, Note on irreducible rings, forthcoming in Proc. Imp. Acad. Japan; T. Nakayama-G. Azumaya, On irreducible rings, Ann. Math. 48 (1947).
4) T. Nakayama, Semilinear normal basis for quasifields, Amer. Journ. Math. 71 (1949).
5) Cf. Remark 6 in § 3.
6) We may make a similar statement for instance for the largest fully reducible two-sided ideals. That (N,_??_) is the radical of (R,_??_) also follows from a weaker assumption that _??_ induces a Galois group in R/N; observe that (R/N,_??_) is then semisimple.
7) Cf. e. g. K. Shoda, Über direkt zerlegbare Gruppen, Journ. Fac. Sci. Tokyo Imp. Univ. Sec. I. Vol. 2 (1930), Satz 3.
8) For the following proof cf. Nakayama, 1. c. 4), § 1. Cf. also Lemma 5 below (in § 3).
9) Cf. Remarks 5, 6 in § 3. (R,_??_) is inverse-isomorphic to (R,_??_) α'u_σ^-1←→u_σα.
10) The (R',_??_)-module R is, by x←→x', isomorphic to the (R',_??_)-(right-) module R' defined similarly as R.
11) For Lemma 4 and Theorem 1 cf. Remark 5 below in § 3.
12) From α), γ) follows, generally, that if m is an R-regular module and R is regular with rank h over its subring T, then the T-endomorphism-ring T^* of m is regular and of rank h with respect to the R-endomorphism-ring R^*.
13) We may weaken the assumption somewhat by decomposing U into directly indecomposable right-ideals and allowing different ranks, so to speak, with respect to non-isomorphic components.
14) Cf. G. Azumaya. On generalized semi-primary rings and Krull-Remark-Schmidt theorem, in Jap. Journ. Math. 19 (1948)
15) In case of infinite rank f=∞ we mean by this that q=∞ and γ is a rational number <g; the restriction being rather inessential.
16) Nakayama, 1. c. 4).
17) Classes are with respect to the subgroup of inner automorphisms. Automorphisms σ in a same class (and only those in case m=R) give isomorphic two-sided moduli (m,σ). So we can speak of (m,σ) with an automorphism-class σ.
18) Cf. Nakayama-Azumaya, 1. c. 3).
19) For the case when _??_ is homomorphic to r, as well as for some remarks on weakening the assumptions, cf. the full proof below at the end of the paper. Cf. also Lemma 2 of the previous note.
20) Our lemmas 1, 2 in §1 remain valid for crossed product with automorphism class-group and Galois class-group.
21) This was pointed out to the writer by G. Azumaya.
22) Cf. K. Shoda, 1, c. 7).
23) It follows from this, combined with Lemma 1, that also the 1-1 correspondence, in Lemma 2, between two-sided ideals of (R,_??_) and _??_-invariant two-sided ideals of R holds, under the weaker assumption that (***) is satisfied for every residue-ring of R modulo a _??_-invariant two-sided ideal

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