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Note on Subdirect Sums of Rings

Published online by Cambridge University Press:  22 January 2016

Masayoshi Nagata
Masayoshi Nagata*
Affiliation:
Mathematical Institute, Nagoya University
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In my previous paper “On the theory of semi-local rings,” we saw that if a semi-local ring R with maximal ideals p1 …,ph is a subdirect sum of local rings Rm,2) then R is the direct sum of R[p1] (proposition 15, (slr)1)) and that a complete semi-local ring is a direct sum of complete local rings (Remark to proposition 5, (slr)).

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 2 , February 1951, pp. 49 - 53
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1951

References

1) To appear in Proc, Jap. Acad, and will be referred as (slr) in the present note.

2) This notation is same as in (slr); this denotes the topological quotients ring of pi with respect to R: See Chapter I, (slr).

3) A ring means an associative ring.

4) An ideal means a two-sided ideal.

5) Since R2 = R, any maximal ideal is prime (we say an ideal p in a ring R is maximal if R ≠ p and if there exists no ideal a such as R ⊃ a ⊃ p)

6) Set theoretical union.

7) Evidently this number is finite.