Is this Corollary a historical or modern product?

Question

I saw this Corollary on page 21 of this book "Introduction to commutative algebra" by Atiyah and Macdonald.

Corollary 2.5. Let $M$ be a finitely generated $A$-module and let $a$ be an ideal of $A$ such that $aM=M$. Then there exists $x \equiv 1 \pmod {a}$ such that $xM=0$

However, I think this acts like a stepping stone from the generalized version of the Cayley-Hamilton theorem to the Nakayama lemma. But I'm curious how this Corollary came about. Did this corollary appear in a historical development process, or is it a corollary that was put in to make it easier to understand from a modern perspective?

Answer 1

Jacobson's 1945 paper, "The Radical and Semi-Simplicity for Arbitrary Rings", has

Theorem 10. If $\mathfrak{N}$ is a right ideal with a finite basis contained in the radical $\mathfrak{R}$ then either $\mathfrak{N}\mathfrak{R} < \mathfrak{N}$ or $\mathfrak{N} =0$

Azumaya's 1951 "On Maximally Central Algebras" goes

Following N. Jacobson$^{7)}$ the radical $N$ of $R$ is defined to be the join of all quasi-regular right ideals of $R$. [...]

Theorem 1. Let $\mathfrak{M}$ be a finitely generated $R$-right-module such that $\mathfrak{M}N = \mathfrak{M}$. Then necessarily $\mathfrak{M} = 0$

The proof is virtually the same as that of Jacobson [9], Theorem 10, [...]

From now on, we assume that $K$ is a commutative ring with unit element and when we deal with moduli with operator ring $K$ we assume always that the unit element of $K$ operates as an identity endomorphism.

Theorem 5. Let $\mathfrak{M}$ be a finite $K$-module such that $\mathfrak{p}\mathfrak{M}=\mathfrak{M}$ for every maximal ideal $\mathfrak{p}$ of $K$. Then we have $\mathfrak{M} = 0$

and Nakayama's 1951 "A Remark on Finitely Generated Modules" follows

Theorem 5 of Azumaya's recent article$^{1)}$ can be formulated in the following generalized form:

I. Let $R$ be a ring. Let $\mathfrak{m}$ be a finitely generated right-module of $R$ such that $\mathfrak{m}R = \mathfrak{m}$. Assume that $\mathfrak{m} = u_1r+u_2r+...+u_mr$ for every generating system $u_1+u_2+...+u_m$ of $\mathfrak{m}$ and for every maximal right-ideal $r$ of $R$. Then $\mathfrak{m} = 0$ [...]

It is perhaps of interest to observe that from this generalization Jacobson's theorem$^{2)}$ may be derived:

II. Let $R$ be a ring and $N$ be its radical. If $\mathfrak{m}$ is a finitely generated right-module of $R$ and if $\mathfrak{m} = \mathfrak{m}N$, then $\mathfrak{m} = 0$ [...]

III. In I we may restrict ourselves to those maximal right-ideals $r$ which contain the radical $N$ [...]

IV. If $M$ is a two-sided ideal and if $\mathfrak{m} = 0$ is the only finitely generated module with $\mathfrak{m} = \mathfrak{m}M$, then $M \subseteq N$

then, Jacobson's "Lectures in abstract algebra" have no reference for either 'Krull' (besides the Krull-Schmidt theorem), 'Azumaya', or 'Nakayama', and only treat Cayley-Hamilton for matrices; no 'Nakayama' in Zariski-Samuel too

the form you quote appears in Bourbaki's "Commutative algebra", as 'Corollary 3.' in II.2.2; they use

THEOREM 2 ("Nakayama's lemma"). - Let $M$ be an $A$-module and $\mathfrak{a}$ a two-sided ideal of $A$. Suppose that one of the following conditions is satisfied:

(i) The $A$-module $M$ is finitely generated, and $\mathfrak{a}$ is contained in the radical of $A$.

(ii) The ideal $\mathfrak{a}$ is nilpontent.

If $N$ is a submodule of $M$ such that $M = N + \mathfrak{a}M$, then we have $N = M$. In particular, if the module $M$ is nonzero, then we have $M \neq \mathfrak{a}M$

from their "Algebra", VIII.9.3 as the last step in the proof; the generalized Cayley-Hamilton is "Proposition 20" also on "Algebra", III.8.11, relatively isolated; fact, they appear to occur isolated from one another generally

the same form you quote also appears on Matsumura's 1986 "Commutative Ring Theory" as 'Theorem 2.2', this time after the generalized Cayley-Hamilton;

the main point, I believe, linking this form to the earlier ones involving radicals is precisely the lemma Bourbaki uses to prove 'Corollary 3':

LEMMA 1. For every ideal $\mathfrak{a}$ of $A$, the set $S$ of elements $1+a$, where $a \in \mathfrak{a}$, is a multiplicative subset of $A$ and the set $\mathfrak{a}'$ of elements of $S^{-1}A$ of the form $a/s$, where $a \in \mathfrak{a}$ and $s \in S$, is an ideal contained in the Jacobson radical of $S^{-1}A$