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Corrections and Supplementaries to My Paper concerning Krull-Remak-Schmidt’s Theorem

Published online by Cambridge University Press:  22 January 2016

Gorô Azumaya
Gorô Azumaya*
Affiliation:
Department of Mathematics, Nagoya University
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It has recently been found that my previous paper “On generalized semi-primary rings and Krull-Remak-Schmidt’s theorem” Jap. Journ. Math. 19 (1949) — referred to as S. K. — contained in its Theorems 19 and 20 some errors. Nevertheless the writer has been able to correct them in suitable forms so that most parts of both theorems hold, even under a weaker assumption, and also subsequent theorems remain valid. These will be, together with some supplementary remarks, shown in the present note.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 1 , June 1950 , pp. 117 - 124
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1950

References

1) We take this opportunity to correct the following errata: in the sixth line following the proof of Theorem 16 (page 537) both should be replaced by .

2) S. K. Lemma 6.

3) Cf. Corollary 1 to Theorem 15; there the assumption that the unit element is the only non-zero idempotent element is superfluous; it follows automatically.

4) = condition B) in S. K. Theorem 19.

5) = non-automorphisms = non-regular elements in the endomorphism ring of mμ.

6) This theorem can readily be transferred, together with Lemma 1, to non-commutative groups if we consider “normal endomorphisms” and make use of the notion of their “Addierbarkeit.” Cf, footnote (27) in S. K.

7) The validity of this lemma and that of the succeeding theorem under the assumption (*) alone were communicated to the writer by T. Nakayama.

8) Generally, if m is a directly indecomposable module satisfying the same assumption as (*) and if {a 0} be a summable system of proper endomorphisms of m then the sum is also proper.

9) This theorem was given in S. K. Theorem 20, i) (in its proof, to be precise) and iii) by assuming not only the assumption (*) but also the assumption (**) below.

10) Cf. Corollary to S. K. Theorem 11.

11) Take account of footnote 8).

12) Because for an arbitrary non-zero element u of m μ ue μ ae v vanishes for almost all v.

13) Cf. S. K. Theorem 8.

14) Cf. S. K. Lemma 8.

15) = condition A) in S. K. Theorem 19.

16) But in case M is finite a is always an automorphism, as Theorem 2, ii) shows.

17) We may take R 0 to be a valution ring of a p-adic number field, for instance.

18) Cf. S. K. Theorem 22.