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A Remark on Finitely Generated Modules, II

Published online by Cambridge University Press:  22 January 2016

Tadasi Nakayama
Tadasi Nakayama*
Affiliation:
Nagoya University and University of Hamburg
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The following small remark on the relationship of the (quasi-)regularity and the zero-divisor property may be more or less known but does not seem to the writer to have been explicitly stated in a literature.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 9 , October 1955 , pp. 21 - 23
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1955

References

1) A remark on finitely generated modules, Nagoya Math. J. 3 (1951), 139-140. Thus, we have merely to consider with a maximal right-ideal with left modulo-unit in which r is not contained.

2) S. foot note 1).

3) The line “the family of right-ideals of III,” before the proposition V of the first note, should read “the family of all maximal right-ideals with modulo-unit.”

Further, the writer was perhaps too hasty when he wrote, in the proof to the proposition V there, that the implication of (B) from (A) was “evident.” The “evident” needs an explanation. It is indeed evident if we take into account (the proposition 1 there and) the fact that for every maximal right-ideal τ* of the ring R* = R ⊕ Z as above (and as there) the intersection r* ∩ R is either R itself or a maximal right-ideal of R with left modulo-unit; this fact can readily be seen from that if r* ∩ R ≠ R then r* + R = R* whence there is an element c ∈ R with 1- c ∈ r* (whence x ≡ cx mod r* ∩ R for all x ∈ R).

However, what is perhaps better is to prove the implication directly, and the proof is merely to repeat the argument of our proof to the proposition 1 of the first note. Thus, assume (A). From m = mR we have m = u1R +… +unR, for any generating system u1 , …, un of R. So u1 can be expressed in a form u1 = u1c1 + … + uncn (ci ∈ R). Let x1 be the right-ideal of R consisting of elements x such that u1x ∈ u2R + … + unR; in case m = 1 the void sum in the right-hand side stands for 0. As u1x = u1c1x + u2c2x + … + uncnx, whence x — c1x ∈ x1 for any element x of R, c1 is a left modulo-unit for x1 Suppose here c1 ∉ xv i.e. r1 ≠ R Then there is a maximal right-ideal r0 of R containing r1 , which evidently possesses c1 as a left modulo-unit. By our assumption we have m = u1x 0 + … + unx 0, whence much the more m = u1x0 + u2R + … + unR. Expressing u1c1 accordingly in a form u1c1 = u1a + … with a ∈ r0, we have c1 — a ∈ r1 (a ∈ x0). This is however a contradiction, since r1 r0, c1 ∉ x0. Thus necessarily r1 = R, or, what is the same, u1R u2R + … + unR. Now our assertion m = 0 can be obtained by an easy induction on the minimum number of generating elements.

4) Observe that N is contained in (and coincides with, in fact) the radical of R*.