van der Waerden and the Ascending Chain Condition

A truism in mathematics, and perhaps most subjects, is that its disseminators are as important as its creators. Niccolò Tartaglia (1499-1557) comes to mind, a great and prolific encyclopedist whose works were consulted in his own day and for generations to come. Perhaps the premiere example is Frans van Schooten and his associates, who did so much to organize and present Descartes' deep but scattered and somewhat opaque geometric approach in La Geometrie. van Schooten's "appendices" amplified La Geometrie massively, the 1659-61 edition becoming a textbook for Leibnitz and Newton and the rock that calculus was built on. Of course the disseminators are mathematicians too, among the select few who expand the new path as well as trod it ("Cartesian" coordinates appear nowhere in Descartes, for example, but were introduced by his acolytes). They cast a great net, subsuming existing mathematics into the new framework as they organize and simplify, all the while spreading the word and making it available to a broad audience of practitioners, many having already worked obscurely to clear patches of brambles and more than ready for the big breakthrough.

B. L. van der Waerden was one of the great disseminators of algebra in the twentieth century, consolidating then recent advances in his two-volume work Moderne Algebra, based on lectures of Emmy Noether at Göttingen and Emil Artin at Hamburg and published in German in 1930-31, English translations following in 1949-1950 as Modern Algebra.^[1] Leading algebraist Saunders Mac Lane paid tribute to the book in 1997, citing it as "the most influential text of algebra of the twentieth century".^[2] K.-H. Schlote puts it this way:

van derWaerden’s book effected a fundamental change in algebra and revolutionized mathematicians’ perception of algebraic problems. It opened up a ‘new world’, as he had himself experienced when he came into contact with the new ideas of Göttingen in 1924. Other mathematicians said that after the appearance of the book the mathematical world was different from what it had been before. Algebra and algebraic problems seemed suddenly to take up a central position in mathematical research, and were no longer regarded as problems of peripheral interest ... The success of Moderne Algebra comes down finally to its lively style of presentation. Because of the abstract basis, the presentation of the individual theories is shorter and clearer, and thus has a stimulating effect on many readers.^[3]

Note the use of the present tense in the final clause: "has a stimulating effect on many readers," English reprints coming out as recently as 1991.

Van der Waerden spoke of "modern algebra", but it is also called "abstract algebra", the term "abstract" having the connotation of distilling the essence of something (from the Latin abstrahere, to draw off from). Think of the parallel movement of abstract art also flourishing early in the twentieth century. In mathematics, abstraction is closely tied to axiomatization and drawing out the underlying concepts and structures subsuming multiple concrete instances. It would be hard to find a better explanation of this phenomenon than van der Waerden's:

The quantities employed in algebraic and arithmetic operations vary in nature; at times we use the integers, or the rational, the real, complex or algebraic numbers, and at other times we deal with polynomials, or rational functions in \( n \) variables, etc. Later on we shall become familiar with quantities of a completely different nature, such as hypercomplex numbers, residue classes, etc., with which we can operate in the same or almost the same manner as with numbers. It is, therefore, desirable to arrive at a common concept embracing all these domains, and to investigate the rules of operation in these domains in general.^[4]

This passage prefaces his discussion of "rings", sets of elements which can be added and multiplied according (mostly) to the conventional rules of normal arithmetic.^[5] Rings needn't be commutative though (a ring is commutative if ab=ba for all elements a,b in the ring). William Rowan Hamilton pioneered non-commutative systems with his Quaternions in 1843, kicking off a rich theory under the term "Hypercomplex Numbers", today called algebras. Wedderburn proved a major structure theorem for algebras in 1908^[5'], a result well-known to van der Waerden and his mentor Emmy Noether — Chapter XVI in Volume II of Moderne Algebra takes up the subject in detail, "Theory of the Hypercomplex Quantities". Another example of the non-commutative tradition is group rings, first discussed by Emmy Noether in 1929 in a paper in which she collaborated with van der Waerden.^[5'']

The prototypical example of a commutative ring is the integers \( \mathbb{Z}, \) illustrating that rings need not have multiplicative inverses (fractions). I'm assuming all rings are commutative and contain a multiplicative identity \( 1 \) in the following, as is the case with the integers.

Other examples of rings are the integers modulo \( n \) for \( n = 2, 3, 4, \ldots. \) These rings are denoted by \( \mathbb{Z} / n \mathbb{Z}, \) where \( n \mathbb{Z} \) consists of all integers that are a multiple of \( n. \) To illustrate, let \( n = 6, \) so \( 6 \mathbb{Z} = \{\ldots -12, -6, 0, 6, 12, \ldots\} \). In this case:
\[ \mathbb{Z} / 6 \mathbb{Z} = \{\overline{0}, \overline{1}, \overline{2}, \overline{3}, \overline{4}, \overline{5}\}, \]
where:

\begin{align}
\overline{0} &= 6 \mathbb{Z} + 0 = \{\ldots -12, -6, 0, 6, 12, \ldots\},\\
\overline{1} &= 6 \mathbb{Z} + 1 = \{\ldots -11, -5, 1, 7, 13, \ldots\},\\
\overline{2} &= 6 \mathbb{Z} + 2 = \{\ldots -10, -4, 2, 8, 14, \ldots\},\\
\overline{3} &= 6 \mathbb{Z} + 3 = \{\ldots -9, -3, 3, 9, 15, \ldots\},\\
\overline{4} &= 6 \mathbb{Z} + 4 = \{\ldots -8, -2, 4, 10, 16, \ldots\},\\
\overline{5} &= 6 \mathbb{Z} + 5 = \{\ldots -7, -1, 5, 11, 17, \ldots\}.\\
\end{align}

What could \( \overline{3} + \overline{4} \) possibly mean, adding these two sets? It turns out that if you take any element of \( \overline{3} \) and add it to any element of \( \overline{4}, \) the sum is always an element of \( \overline{1} \)! It's not hard to prove either using simple facts from number theory; operationally, it's a matter of taking the remainder of \( 3 + 4 = 7 \) after dividing by \( 6 \):

\[ 3 + 4 \hspace{-.6em} {\pmod{6}} = 7 \hspace{-.6em} {\pmod{6}} \equiv 1 \hspace{-.6em} {\pmod{6}}. \]

Similarly \( \overline{3} \times \overline{4} = \overline{0} \) because:

\[ 3 \times 4 \hspace{-.6em} {\pmod{6}} = 12 \hspace{-.6em} {\pmod{6}} \equiv 0 \hspace{-.6em} {\pmod{6}}. \]

This shows that even simple rings can have zero divisors, that is, non zero elements whose product is zero, an impossibility in the integers.

The abstracted principle making this work is that \( n \mathbb{Z} \) is an ideal in \( \mathbb{Z} \), that is, a set that is closed additively and by multiplication by any element of \( \mathbb{Z}: \)

\begin{align}
&\text{ 1) If } a, b \in 6 \mathbb{Z}, \text{ then } a - b \in 6 \mathbb{Z},\\
&\text{ 2) If } r \in \mathbb{Z}, a \in 6 \mathbb{Z}, \text{ then } ra \in 6 \mathbb{Z}.
\end{align}

The passage from \( \mathbb{Z} \) to \( \mathbb{Z} / 6 \mathbb{Z} \) described here can be effected for any ideal in any ring and was common coin to Emmy Noether and her associates well before 1920. van der Waerden carries it out with characteristic elegance, including motivating references to the integers modulo \( n. \) He even uses the notation \( a \equiv b {\pmod{I}} \) to denote \( a - b \in I \) for any ideal \( I \) in a ring. One feels frozen in time reading these sections of Modern Algebra in 2020 — they could have been written yesterday, avant-garde though they were in 1930.^[6]

When introducing ideals in the ring \( \mathfrak{o}, \) van der Waerden writes:

The ideal generated by several elements \( a_1, \ldots, a_n \) may be defined as the totality of all sums of the form \[ \sum r_i a_i + \sum n_j a_j \] (or, if \( \mathfrak{o} \) contains an identity, as \( \sum r_i a_i \)), or as the intersection of all ideals of \( \mathfrak{o} \) containing the elements \( a_1, \ldots, a_n \). The ideal is denoted by \( (a_1, \ldots, a_n), \) and \( a_1, \ldots, a_n \) are said to form an ideal basis.^[7]

The \( n_j \) are integers here and \( \sum n_j a_j \) just means adding up \( n_j \) copies of \( a_j \) (or \( -n_j \) copies if \( n_j \) is negative, attaching a negative sign to the result), an unnecessary caveat if \( \mathfrak{o} \) has a \( 1. \) In volume II, he writes:

We say that the basis condition is valid in a ring \( \mathfrak{o} \) when every ideal in \( \mathfrak{o} \) has a finite basis,^[8]

noting that this condition rules out "unduly complicated" rings and makes possible an investigation of "the extent to which simple laws that are valid in a domain such as the integers may be carried over to more general rings" and that "this condition is satisfied in a great many important cases".

He writes that "it is important that the basis condition is equivalent to the following 'divisor chain condition'."^[9]

Divisor Chain Condition, First Statement

If a chain of ideals \( \mathfrak{a}_1, \mathfrak{a}_2, \mathfrak{a}_3 \ldots \) in \( \mathfrak{o} \) is given and if every \( \mathfrak{a}_{i+1} \) is a proper divisor of \( \mathfrak{a}_i: \) \[ \mathfrak{a}_i \subsetneq \mathfrak{a}_{i+1}, \] the chain breaks off after a finite number of terms.

This is equivalent to

Divisor Chain Condition, Second Statement

If an infinite chain of divisors \( \mathfrak{a}_1, \mathfrak{a}_2, \mathfrak{a}_3 \ldots \) is given: \[ \mathfrak{a}_i \subseteq \mathfrak{a}_{i+1} \] all terms must be equal after a certain \( n: \) \[ \mathfrak{a}_n = \mathfrak{a}_{n+1} = \cdots. \]

The term divisor is confusing at first. It's clear that van der Waerden takes ideal \( \mathfrak{a_{i+1}} \) divides ideal \( \mathfrak{a_i} \) to mean that \( \mathfrak{a}_i \subseteq \mathfrak{a}_{i+1}. \) Translate to the integers to see how this makes sense:

\[ (24) \subseteq (6) \subseteq(2) \text{ in } \mathbb{Z} \]

is synonymous with:

\[ 2 \; | \; 6 \; | \; 24 \]

because every multiple of \( 24 \) is a multiple of \( 6 \) and every multiple of \( 6 \) is a multiple of \( 2. \) It's clear then that the basis condition is valid in \( \mathbb{Z} \) because an ascending chain of ideals corresponds to decreasing positive numbers in \( \mathbb{Z} \) and that can't go on indefinitely.

van der Waerden then proves that the basis condition and the divisor chain condition are equivalent. His proof goes as follows:

Assume the basis condition holds in ring \( \mathfrak{o} \) and consider a chain of ideals:

\[ \mathfrak{a}_1 \subseteq \mathfrak{a}_2 \subseteq \mathfrak{a}_3 \subseteq \mathfrak{a}_4, \ldots. \]

Let \( \mathfrak{v} = \cup_{i=1}^\infty \mathfrak{a}_i. \) It's easy to prove that \( \mathfrak{v} \) is an ideal in \( \mathfrak{o} \) (van der Waerden does!). By assumption, every ideal in \( \mathfrak{o} \) has a finite basis, say:

\[ \mathfrak{v} = \bigcup_{i=1}^\infty \mathfrak{a}_i = (a_1, \ldots, a_r). \]

Each of the \( a_i \) making up the basis is in some \( \mathfrak{a}_{n_i}. \) If \( n \) is the greatest of the \( n_i, \) then all the \( a_i \) are in \( \mathfrak{a}_n \) by the chained nature of the \( \mathfrak{a}_i. \) It follows that \( \mathfrak{v} = \mathfrak{a}_n \) and similarly for the rest of the \( \mathfrak{a}_i \) when \( i > n: \)

\[ \mathfrak{v} = \mathfrak{a_n} = \mathfrak{a_{n+1}} = \mathfrak{a_{n+2}} = \cdots. \tag*{QED.} \]

Assume the divisor chain condition holds in ring \( \mathfrak{o} \) and let \( \mathfrak{a} \) be an ideal in \( \mathfrak{o}. \) Choose any \( a_1 \in \mathfrak{a}. \) If \( \mathfrak{a} = (a_1), \) then \( \mathfrak{a} \) has a basis consisting of the single element \( a_1. \) Otherwise there is some element \( a_2 \) in \( \mathfrak{a} \) that is not in \( (a_1), \) so:
\[ (a_1) \subsetneq (a_1, a_2). \]
If \( \mathfrak{a} = (a_1, a_2), \) then \( \mathfrak{a} \) has a basis consisting of the two elements element \( a_1 \) and \( a_2. \) Otherwise there is some element \( a_3 \) in \( \mathfrak{a} \) that is not in \( (a_1, a_2), \) so:
\[ (a_1) \subsetneq (a_1, a_2) \subsetneq (a_1, a_2, a_3). \]
This process cannot continue indefinitely, otherwise the divisor chain condition would not hold. It follows that there is some finite integer \( r \) such that no element \( a_{r+1} \) can be chosen that is not already in \( (a_1, a_2, a_3, \cdots a_r); \) that is, \( \mathfrak{a} = (a_1, a_2, a_3, \cdots a_r) \) and \( \mathfrak{a} \) has a finite basis. QED.

Today (2020), van der Waerden's "divisor chain condition" is called the ascending chain condition (ACC) and instead of saying "the basis condition is valid", we say that every ideal is finitely generated. That's just terminology though, the concepts then and now are identical and go back to van der Waerden, Emmy Noether, Emil Artin, and their predecessors. Rings satisfying this condition are now called Noetherian, a coinage apparently due to Claude Chevalley in 1943.^[10] van der Waerden has acknowledged that Chapter 12 of Moderne Algebra, where he addressed finite bases / the ACC, is based on Emmy Noether's 1921 paper "Idealtheorie in Ringbereichen"^[11], a debt echoed by Jeremy Gray:

One way to measure Emmy Noether’s achievements in abstract algebra is simply to pick up any advanced undergraduate or graduate text book in the subject. Mathematicians simply do ring theory her way. They may stipulate that a ring has a multiplicative unit, but that’s the only change they make. That is why rings that satisfy an ascending chain condition are called Noetherian rings, in her honour. They form a large class of rings that very neatly and naturally captures what is happening in the rings of interest in number theory and algebraic geometry.^[12]

At the start of Chapter XII in Volume II (General Ideal Theory of Commutative Rings), van der Waerden proves what has come to be known as the Hilbert Basis Theorem:

If the basis condition is valid in the ring \( \mathfrak{o}, \) which contains an identity element, then it is valid in the polynomial domain \( \mathfrak{o}[x]. \)

He notes that the theorem goes back essentially to Hilbert, his proof being due to Artin in a lecture in Hamburg in 1926.^[13] van der Waerden proceeds as follows to prove this theorem.

Let \( \mathfrak{U} \) be an ideal in \( \mathfrak{o}[x] \) — the objective is to prove that \( \mathfrak{U} \) has a finite basis, given that ideals in \( \mathfrak{o} \) do. Any non-zero polynomial \( p(x) \in \mathfrak{o}[x] \) can be written in the form

\begin{align}
p(x) = p_k x^k + p_{k-1} x^{k-1} + \cdots + p_1 x + p_0, \;\;\text{ some } k = 0, 1, 2, \ldots, \;\; p_i \in \mathfrak{o}, \;\; p_k \neq 0,
\end{align}

where \( x^k \) is assumed to be the highest power of \( x \) represented; that is, the coefficients of powers of \( x \) greater than the \(k\)^th are assumed to be zero. Then the degree of \( p(x) \) is \( k \) (\( \deg{p} = k \)) and \( p_k \) is said to be the leading coefficient of \( p(x). \) Let:

\begin{align}
\mathfrak{a} &= \text{ set of leading coefficients of the polynomials in } \mathfrak{U}, \text{ together with } 0.
\end{align}

Then \( \mathfrak{a} \) is an ideal in \( \mathfrak{o}. \) To see this, suppose \( \alpha \in \mathfrak{a}. \) Then \( \alpha \) is the leading coefficient of some \( a(x) \in \mathfrak{U}: \)

\begin{align}
a(x) &= \alpha x^n + \cdots \in \mathfrak{U}, \;\; \text{ some } n = 0, 1, 2, \ldots.\\
\end{align}

Similarly \( \beta \in \mathfrak{a} \) implies that \( \beta \) is the leading coefficient of some \( b(x) \in \mathfrak{U}: \)

\begin{align}
b(x) &= \beta x^m + \cdots \in \mathfrak{U}, \;\; \text{ some } m = 0, 1, 2, \ldots.\\
\end{align}

Assume \( n \geq m \) and consider \( c(x) = a(x) - x^{n-m} b(x). \) Multiplying \( b(x) \) by \( x^{n-m} \) or indeed by any other polynomial keeps the result in \( \mathfrak{U}, \) since it is an ideal, and subtracting that from another element of \( \mathfrak{U}, a(x), \) keeps the final result \( c(x) \) in \( \mathfrak{U} \) also, so

\begin{align}
c(x) &= a(x) - x^{n-m} b(x) \in \mathfrak{U}\\
&= (\alpha x^n + \cdots ) - (\beta x^n + \cdots)\\
&= (\alpha - \beta) x^n + \cdots.
\end{align}

The upshot is that either \( \alpha - \beta = 0 \) or \( \alpha - \beta \) is the coefficient of the highest power of some polynomial in \( \mathfrak{U}. \) In any event, \( \alpha - \beta \in \mathfrak{a} .\) Similarly \( \lambda \alpha \in \mathfrak{a} \) for any \( \lambda \in \mathfrak{o} \) and so \( \mathfrak{a} \) is an ideal in \( \mathfrak{o} \). Since all ideals in \( \mathfrak{o} \) are assumed to have an finite basis:

\[ \mathfrak{a} = (\alpha_1, \alpha_2, \ldots, \alpha_r), \hspace{20pt} \alpha_i \in \mathfrak{o}. \]

Let the corresponding polynomials in \( \mathfrak{U} \) be \( a_1(x), a_2(x), \ldots a_r(x) \) — they will be part of a basis for \( \mathfrak{U}. \)

Step 1: For \(f(x) \in \mathfrak{U}, \) there is some \( f_1(x) \in \mathfrak{o}[x] \) with \( f(x) - f_1(x) \in (a_1(x), \ldots, a_r(x)), \) where deg \( f_1(x) < n \) for some fixed \( n. \)

To illustrate choosing an \( f_1(x) \) with degree at least one less than that of \( f(x), \) consider:

\begin{align}
\mathfrak{a} &= (\alpha_1, \alpha_2),\\
a_1(x) &= \alpha_1 x^2 + \cdots,\\
a_2(x) &= \alpha_2 x^3 + \cdots,\\
f(x) &= \alpha x^4 + \cdots.
\end{align}

Since \( \alpha \in (\alpha_1, \alpha_2): \)

\begin{align}
\alpha = \lambda_1 \alpha_1 + \lambda_2 \alpha_2, \hspace{20pt} \lambda_1, \lambda_2 \in \mathfrak{o}.
\end{align}

Put:

\begin{align}
f_1(x) &= f(x) - (\lambda_1 x^2 a_1(x) + \lambda_2 x a_2(x))\\
&= (\alpha x^4 + \cdots) - \left(\lambda_1 x^2 (\alpha_1 x^2 + \cdots) + \lambda_2 x (\alpha_2 x^3 + \cdots)\right)\\
&= \bigl(\underbrace{\alpha - (\lambda_1 \alpha_1 + \lambda_2 \alpha_2)\bigl)}_{ \Large 0 }x^4 + \text{ terms in lower powers of } x.\\
\end{align}

The degree of \(f_1(x) \) is at least one less than the degree of \(f(x) \) and \( f(x) - f_1(x) \in (a_1(x), a_2(x)) \) because it is a linear combination of the \( a_i(x) \) by constants and powers of \( x, \) the latter allowable when dealing with ideals in \( \mathfrak{o}[x]. \) The example carries the essence of the reduction no matter how many \( a_i \) there are, so it is always possible to replace \( f(x) \) by a polynomial of lower degree modulo \( (a_1(x), a_2(x), \cdots, a_r(x)) \) in this fashion.

The example produced a polynomial of degree three or less and could have been repeated to produce a polynomial of degree two or less, the limiting factor being the greatest degree of the \( a_i(x). \) In the general case, the process can be continued to produce an \( f_1(x) \) of degree less than \( n = \max( \text {deg } a_i(x)). \) This completes step 1.

Let
\begin{align}
\mathfrak{a}_{n-1} &= \text{ set of coefficients of } x^{n-1} \text{ in the polynomials of degree } \leq n-1 \text{ in } \mathfrak{U}, \text{ together with }0\\
&= \{\beta \in \mathfrak{o} \; | \; b(x) = \beta x^m + \cdots \text{ for some } b(x) \in \mathfrak{U} \text { with } m \leq n-1\} \cup \{0\}.
\end{align}

It's easy to show that \( \mathfrak{a}_{n-1} \) is an ideal in \( \mathfrak{o}, \) so it has a finite basis, say \( (\alpha_{r+1}, \cdots, \alpha_s). \) Assume that the corresponding polynomials in \( \mathfrak{U} \) are \( a_{r+1}(x), \cdots, a_s(x), \) where:

\begin{align}
a_{r+i}(x) = \alpha_{r+i} x^{n-1} + \cdots.
\end{align}

Step 2: For \(g(x) \in \mathfrak{U} \) with deg \( g(x) \leq n-1, \) there is some \( g_1(x) \in \mathfrak{o}[x] \) with \( g(x) - g_1(x) \in (a_{r+1}(x), \ldots, a_s(x)), \) where deg \( g_1(x) \leq n-2 \).

As in step 1, it is only a matter of subtracting a suitable linear combination of the \( a_i(x), \) but even simpler because in this case it's not necessary to weight with powers of \( x \). This completes step 2.

Combining these two steps, every polynomial of \( \mathfrak{U} \) is congruent to a polynomial of degree \( \leq n-2 \) modulo \( (a_1(x), \ldots, a_r(x), a_{r+1}(x), \ldots, a_s(x)). \) Repeatedly execute step 2, reducing the degree by one at each step at the cost of introducing a new batch of polynomials into the basis. Finally only constants are left and every polynomial of \( \mathfrak{U} \) is congruent to zero modulo:

\begin{align}
(a_1(x), \ldots, a_r(x), a_{r+1}(x), \ldots, a_s(x), \ldots, a_{v+1}(x), \ldots, a_w(x)).
\end{align}

In other words, the polynomials \( a_1(x), \ldots a_w(x) \) form a basis for the ideal \( \mathfrak{U} \) in \( \mathfrak{o}[x]. \) Since this is true of any ideal in \( \mathfrak{o}[x], \) the basis condition is satisfied in \( \mathfrak{o}[x]. \) QED.

Even the mighty French Revolution, perhaps the most thoroughgoing top-to-bottom social reorientation of modern times, proceeded in stages: storming the Bastille, total mobilization for defense, executing the king, the Directory, and so on. All the same, such moments of sharp clarity are extraordinarily rare. Emmy Noether, Emil Artin, and others had begun the change in earnest in the world of mathematics, aspiring Robespierres in the provinces^[14], but Moderne Algebra amounted to storming the Bastille, the decisive step announcing the new way to the world and from which there was no turning back.

Of course the new, abstract approach was based on what had gone before — the Hilbert Basis Theorem, discussed above at length, was first proven by David Hilbert in a recognizable form in 1890 after all. The difference is that abstraction became not an approach, but the approach to algebra after van der Waerden was done. The reigning algebra textbook in Germany before Moderne Algebra was Heinrich Weber's Lehrbuch der Algebra, first published in 1895 and going through many reprints. Weber had himself delved deeply into polynomial theory and was conversant with the new approach, yet insisted that algebra is the discipline whose aim is the resolution of equations, an approach guiding his textbook. Moderne Algebra overthrew the old way once and for all, broadcasting the structural, abstract approach to a receptive community of researchers. Let Leo Corry have the last word:

The main difference between two textbooks [Lehrbuch der Algebra and Moderne Algebra] lies not only in the bodies of knowledge they put forward, but especially in their images of knowledge. The difference is manifest, in particular, in the fact that van der Waerden turned the conceptual hierarchy of algebra upside down, by introducing groups, fields and many other concepts in all their generality from the outset. The fact that Weber did not adopt the abstract approach in his own Lehrbuch attests for the fact that between the dates of publication of this book and of van der Waerden's Moderne Algebra, the general conception of algebra had fundamentally changed. This change, however, was not brought about by the mere adoption of the abstract formulation in algebra nor by the steady growth of the body of knowledge. Rather, it was the product of a deeper, overall transformation of the aims and methods of algebra.^[15]

Mike Bertrand

February 13, 2020

---

^ 1. Moderne Algebra. Unter Benutzung der Vorlesungen von E. Artin und E. Noether, by B. L. van der Waerden, Verlag Julius Springer (2 volumes, 1930-1931). English translation: Modern Algebra. In part a development from lectures by E. Artin and E. Noether (translated from the 2nd German edition by Fred Blum (Vol. I) and Theodore J. Benac (Vol II), with revisions and additions by the author), Ungar (1949-1950).

^ 2. "Van der Waerden's Modern Algebra", by Saunders Mac Lane, Notices of the American Mathematical Society, 44 3 (1997), pp. 321-322.

^ 3. "B. L. van der Waerden, Moderne Algebra, First Edition (1930-1931)", by K.-H Schlote, pp.901-916 in Landmark Writings in Western Mathematics, 1640-1940, edited by I. Grattan-Guiness, Elsevier Science (2005), ISBN 978-0471963172. This valuable article summarizes the entire technical content of Moderne Algebra in addition to assessing its impact.

^ 4. See Modern Algebra, Volume I, Chapter III, § 11, "Rings", p. 32. This and all subsequent citations are to the 1949-1950 English editions. The first axiomatic definition of a ring was given by Abraham Fraenkel in 1914, short years before Moderne Algebra was published (see A History of Abstract Algebra, by Israel Kleiner, Birkhäuser (2007), ISBN 978-0-8176-4684-4, p. 58).

^ 5. Modern Algebra, Volume I, Chapter III, § 11, "Rings", pp. 32-38.

^ 5'. "On Hypercomplex Numbers", by J. H. M. Wedderburn, Proceedings of the London Mathematical Society, 6 (1908), pp. 77–118.

^ 5''. "Hyperkomplexe Größen und Darstellungstheorie" ("Hypercomplex quantities and representation theory"), by Emmy Noether, Math. Zeitschr., 30 (1929), pp. 641–692. See Peter Roquette's "Emmy Noether’s contributions to the theory of group rings", 2002.

^ 6. Modern Algebra, Volume I, Chapter III, § 16, "Ideals. Residue Class Rings", pp. 49-53.

^ 7. Modern Algebra, Volume I, p. 50.

^ 8. Modern Algebra, Volume II, Chapter XII, § 84, "Basis Condition and Divisor Chain Condition", p. 18.

^ 9. Modern Algebra, Volume II, p. 21.

^ 10. See “On the Theory of Local Rings,” by Claude Chevalley, Annals of Mathematics, 44 (1943), pp. 690-708. The citation is in Jeff Miller's invaluable Earliest Known Uses of Some of the Words of Mathematics.

^ 11. See "On the Sources of My Book Moderne Algebra", by B. L. van der Waerden, Historia Mathematica, 2 (1975), pp. 31-40. The pivotal paper is "Idealtheorie in Ringbereichen" (Ideal Theory in Ring Domains), by Emmy Noether, Mathematische Annalen, 83 (1) (1921), pp 24–66. The second link is Daniel Berlyne's English translation from 2014.

^ 12. A History of Abstract Algebra: From Algebraic Equations to Modern Algebra, by Jeremy Gray, Springer (2018), ISBN 978-3-319-94772-3, p. 294.

^ 13. See Modern Algebra, Volume II, pp. 18-20 for the Hilbert Basis Theorem. Hilbert's original paper on this subject is "Ueber die Theorie der algebraischen Formen", Mathematische Annalen, 36 (4) (1890), pp. 473–534. Jeremy Gray translated the pertinent part of this paper into English and discussed it helpfully in Chapter 25 of History of Abstract Algebra, pp. 263-273. This is the theorem about which Paul Gordan reportedly exclaimed, “This is not mathematics, it is theology!” (Gray, p. 266). van der Waerden credits Artin for the proof he gives of the Basis Theorem in "Sources" (1975).

^ 14. As a radical socialist, pacifist, and friend of the Soviet Union, Emmy Noether might have appreciated this comparison. See "In Memory of Emmy Noether", by P. S. Alexandroff, pp. 99-111 in Emmy Noether: A Tribute to Her Life and Work, edited by James K. Brewer and Martha K. Smith, Marcel Dekker (1981), ISBN 0-8247-1550-0. Colin McLarty confirms and amplifies Alexandroff's picture in "Poor Taste as a Bright Character Trait: Emmy Noether and the Independent Social Democratic Party", Science in Context 18(3) (2005), pp. 429–450. The "poor taste" in the title refers to the attitude of some of Noether's colleagues to her political stance, according to Alexandroff, which he regarded as a "bright character trait".

^ 15. Modern Algebra and the Rise of Mathematical Structures, 2nd revised edition, by Leo Corry, Birkhäuser (2004), ISBN 3-7643-7002-5, p. 63. The first edition of this invaluable book was published in 1996. He compares Weber's Lehrbuch der Algebra and Moderne Algebra at length and writes of a "turning point" (p. 20). See in particular Chapter 1 (pp. 19-63).