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 +… +u_nR, for any generating system u₁ , …, u_n of R. So u₁ can be expressed in a form u₁ = u₁c₁ + … + u_nc_n (c_i ∈ R). Let x₁ be the right-ideal of R consisting of elements x such that u₁x ∈ u₂R + … + u_nR; in case m = 1 the void sum in the right-hand side stands for 0. As u₁x = u₁c₁x + u₂c₂x + … + u_nc_nx, whence x — c₁x ∈ x₁ for any element x of R, c₁ is a left modulo-unit for x₁ Suppose here c₁ ∉ x_v 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 + … + u_nx ₀, whence much the more m = u₁x₀ + u₂R + … + u_nR. 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 + … + u_nR. 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*.