8. Non commutative algebra.

We have seen (p. 62) how the first non commutative algebras made their appearance in 1843-44, in the work of Hamilton [145 a] and of Grassmann ([134], v. I_2). Hamilton, in introducing the quaternions, already has a very clear conception of arbitrary algebras of finite rank over the field of real numbers ([145 a], Preface, pp. (26)-(31)).$^1$ In developing his theory, he has, a little later, the idea of considering what he calls “biquaternions”, that is to say the algebra over the complex numbers having the same multiplication table as the field of quaternions; and he observes on this occasion that this extension has the effect of provoking the apparition of divisors of zero ([145 a], p. 650). The point of view of Grassmann is somewhat different, and for a long time his “exterior algebra” will remain more or less apart from the theory of algebras,$^2$ but often in his language, which still lacks precision, the first idea of an algebra (of finite dimension or not) defined by a system of generators and relations cannot miss being recognised ([134], v. II_1, pp. 199-217).

New examples of algebras are introduced in the years 1850-1860, in a more or less explicit way: if Cayley, developing the theory of matrices ([58], v. II, pp. 475-496), does not consider yet that square matrices make up an algebra (a point of view that will not be expressed clearly until the Peirces around 1870 [248 c]), at least he notes already, on this occasion, the existence of a system of matrices of order 2 which satisfy the multiplication table of the quaternions, a remark that can be considered as the first example of a

$^1$ The concept of the isomorphism of two algebras is not mentioned by Hamilton; but from this time mathematicians of the English school, and notably of Morgan and Cayley, know well that a change of basis does not modify substantially the algebra being studied (see for example the work of Cayley on algebras of rank 2 ([58], v. I, pp. 128-130)).

$^2$ Perhaps the reason for this must be seen in the fact that apart from the “exterior” multiplication, Grassmann introduces also between the multivectors what he calls the “regressive” and “interior” multiplications (which take the place for him of all that touches on duality). It is in any case fairly remarkable that, around 1900 still, in the article Study-Cartan of the Encyclopaedia ([52 a], v. II_1, pp. 107-246), the exterior algebra is not classed among the associative algebras, but receives separate treatment, and that it is not pointed out that one of the types of algebras of rank 4 (type VIII of p. 180) is none other than the exterior algebra over a space of dimension 2.

linear representation of an algebra.$^3$ On the other hand, in the memoir where he defines the abstract notion of a finite group, he also gives in passing the definition of the algebra of such a group, without besides getting anything out of this definition ([58], v. II, p. 129).

There is no other notable progress to point out before 1870; but at that time research on the general structure of algebras of finite dimension (over the field of real or complex numbers) starts. It is B. Peirce who takes the first steps along this way; he introduces the notions of nilpotent element, of idempotent element, proves that an algebra (with or without identity) of which at least one element is not nilpotent contains an idempotent $\neq 0$, writes down the famous decomposition

(e idempotent, $x$ an arbitrary element), and has the idea (still a bit imprecise) of the decomposition of an idempotent into a sum of "primitive" idempotents orthogonal in pairs [247]. As well, according to Clifford ([65], p.274),$^4$ it is to B. Peirce that must be attributed the notion of the tensor product of two algebras, that Clifford himself applies implicitly to a generalisation of the "biquaternions" of Hamilton ([65], pp. 181-200), and explicitly to the study of the algebras that bear his name, a few years later ([65], pp. 397-401 and 266-276). These new notions are used by B. Peirce for the classification of algebras of small dimension (over the complex numbers), a problem which is also attacked, around 1880, by other mathematicians of the anglo-American school, Cayley and Sylvester in the lead. The great variety of possible structures thus becomes rapidly obvious, and it is no doubt this fact that, during the following period, is going to direct research towards obtaining classes of algebras with more particular properties.

On the continent, where the evolution of ideas is fairly different, such research appears already before 1880. In 1878, Frobenius proves that the quaternions constitute the only example of a non commutative field (of finite dimension) over the field of real numbers ([119], v. I, pp. 343-405) — a result published independently two years later by C. S. Peirce [248 d]. From 1861, Weierstrass, stating accurately a remark of Gauss, had, in his lectures,

$^3$ Truth to say, Cayley does not prove this existence, does not write down explicitly the matrices in question, and does not seem to have noticed at that moment that certain ones are necessarily imaginary (in all this memoir, it is never made clear if the "quantities" that occur in the matrices are real or complex; there occurs incidentally however a complex number in p. 494). One would think that there is only one more step needed in order to identify the "biquaternions" of Hamilton with the complex matrices of order 2; in fact, this result will not be explicitly stated until the Peirces in 1870 ([247], p. 132). The general idea of a regular representation of an algebra is introduced by C. S. Peirce around 1879 [248 c]; it was foreseen by Laguerre already in 1876 ([192], v. I, p. 235).

$^4$ B. Peirce met Clifford in London in 1871, and both the one and the other allude many times to their conversations, of which one took place no doubt at the meeting of the London Mathematical Society, where Peirce had presented his results.

characterised commutative algebras without nilpotent elements$^5$ over $\mathbf{R}$ or $\mathbf{C}$ as direct sums of fields (isomorphic to $\mathbf{R}$ or $\mathbf{C}$); Dedekind had on his side arrived at the same conclusions around 1870, in connection with his "hypercomplex" conception of the theory of commutative fields; their proofs were published in 1884-85 ([329 a], v. II, pp. 311-332 and [79], v. II, pp. 1-19). It is in 1884 also that H. Poincaré, in a short very cryptic note ([251 a], v. V, pp. 77-79), draws attention to the possibility of considering the equations $z_i = \phi_i(x_1, \ldots, x_n, y_1, \ldots, y_n)$ which express the multiplicative law $(\sum_i x_i e_i)(\sum_i y_i e_i) = \sum_i z_i e_i$ in an algebra, as defining (locally of course) a Lie group. This remark seems to have made a big impression on Lie and his disciples (Study, Scheffers, F. Schur and a little later Molien and E. Cartan), busy precisely at this time with the development of the theory of "continuous" groups, and notably the problems of classification (see in particular [271], p. 387); during the period 1885-1905, it guides the mathematicians of this school to apply to the study of the structure of algebras methods of the same kind as those used by them in the study of Lie groups and Lie algebras. These methods rely above all on the consideration of the characteristic polynomial of an element of the algebra relative to its regular representation (a polynomial already met in the work of Weierstrass and Dedekind quoted earlier) and on the decomposition of this polynomial into irreducible factors; a decomposition where, as Frobenius will discover a bit later, is reflected the decomposition of the regular representation into irreducible components.

During the research of the school of Lie on algebras the "intrinsic" notions of the theory gradually emerge. The notion of radical appears in a particular case (that where the quotient by the radical is a direct summand of the field) with G. Scheffers in 1891 [271], more clearly with Molien [224 a] and Cartan ([52 a], v. II$_1$, pp. 7-105), who study the general case (the word "radical" itself is from Frobenius ([119] v. III, pp. 284-329)). Study and Scheffers [271] put into prominence the concept of an algebra as a direct sum of several others (already foreseen by B. Peirce ([247], p. 221)). Finally with Molien [224 a] quotient algebras of an algebra are introduced, a notion that is essentially equivalent to that of a twosided ideal (defined for the first time by Cartan ([52 a], v. II$_1$, pp. 7-105)) or of homomorphism (a name also due to Frobenius); the analogy with groups is very sharp here, and a little later in 1904, Epsteen and Wedderburn consider composition sequences of twosided ideals and extend to them the Jordan-Hölder theorem. The most important results of this period are those of T. Molien [224 a]: guided by the notion of a simple group, he defines simple algebras (over $\mathbf{C}$) and proves that these are matrix algebras, then proves that the structure of an arbitrary algebra of finite rank over

$^5$ In fact, Weierstrass imposes on his algebras a stricter condition, namely that the equation

C reduces essentially to the case (already studied by Scheffers) where the quotient by the radical is a direct sum of fields. These results are soon after found again and established in a clearer and more rigorous way by E. Cartan ([52 a], v. II₁, pp. 7-105), who introduces on this occasion the notion of semi-simple algebra, and brings out numerical invariants (the “Cartan integers”) linked with an arbitrary algebra over the field C – thus bringing the theory of these algebras to a point beyond which we have hardly progressed since;$^6$ finally he extends the results of Molien and his own to algebras over $\mathbf{R}$.

Around 1900 a movement of ideas grows up which leads to the abandonment of all restrictions on the field of scalars in everything touching on linear algebra; in particular we must bring out the vigorous impulsion given to the study of finite fields by the American school, around E. H. Moore and L. E. Dickson; the most striking result of this research is Wedderburn’s theorem [328 a] proving that every finite field is commutative. In 1907, Wedderburn picks up again Cartan’s results and extends them to an arbitrary base field [328 b]; doing this he abandons completely the methods of his forebears (which become inapplicable as soon as the base field is no longer algebraically closed or totally ordered), and returns, by perfecting it, to B. Peirce’s technique of idempotents, which allows him to put into definitive form the theorem on the structure of semi-simple algebras, whose study is reduced to that of non commutative fields. Further, the problem of the extension of the field of scalars is stated naturally in the perspective which he adopts, and he proves that every semi-simple algebra remains semi-simple after a separable extension of the base field,$^7$ and becomes a direct sum of central algebras of matrices if this extension is taken large enough ([328 b], p. 102).$^8$ A little later, Dickson, for $n = 3$ [88 a] and Wedderburn himself for arbitrary $n$ [328 c] give the first examples of non commutative fields of

$^6$ The essential difficulties come from study of the radical, for whose structure no satisfying classification principle has yet been found. $^7$ At the time when Wedderburn was writing, the notion of separable extension had not yet been defined; but he uses implicitly the hypothesis that, if an irreducible polynomial $f$ over the base field has a root $x$ in an extension of this field, then one necessarily has $f'(x) \neq 0$ ([328 b], p. 103). It is only in 1929 that E. Noether pointed out the phenomena linked to the inseparability of the extension of the field of scalars [236 c]. Let us mention here another result linked to questions of separability (and now attached to homological algebra), the decomposition of an algebra as the direct sum (but not direct product!) of its radical and a semi-simple algebra. This result (which had been proved by Molien when the field of scalars is $\mathbf{C}$ and by Cartan for algebras over $\mathbf{R}$) is stated in its general form by Wedderburn, who only proves it in fact when the quotient of the algebra by its radical is simple ([328 b], pp. 105-109) by using besides the same hypothesis on irreducible polynomials as above. $^8$ The arithmetical researches on linear representations of groups, which start at the same time, also lead to the consideration of the equivalent notion of a neutralising field of a representation [279 d].

rank $n^2$ over their centre,$^9$ inaugurating thus in a particular case the theory of "crossed products" and of "factor sets" that would be developed later by R. Brauer [34 a] and E. Noether [236 c]. Finally, in 1921, Wedderburn proves a particular case of the commutation theorem [328 d].

In between times, from 1896 to 1910, there was developing, in the hands of Frobenius, Burnside and I. Schur, a neighbouring theory to that of algebras, the theory of the linear representation of groups (limited at first to representations of finite groups). It takes its origins from remarks of Dedekind: this latter (even before the publication of his work on algebras) had, around 1880, met during his research on normal bases of Galois extensions, the "Gruppendeterminant" $\det(x_{st^{-1}})$, where $(x_s)_{s \in G}$ is a sequence of indeterminates for which the set of indices is a finite group $G$ (in other words, the norm of the generic element of the group algebra $G$ relative to its regular representation); and he had observed that when $G$ is abelian, this polynomial decomposes into linear factors (which generalised an identity proved a long time before for "circulant" determinants, which correspond to cyclic groups $G$). During his very interesting correspondence with Frobenius ([79], v. II, pp. 414-442), Dedekind, in 1896, draws his attention to this property, its link with the theory of characters of abelian groups [327 c] and some analogous results on particular non commutative groups, that he had obtained in 1886. A few months later, Frobenius solved completely the problem of the decomposition of the "Gruppendeterminant". into irreducible factors ([119], v. III, pp. 38-77), thanks to his brilliant generalisation of the notion of character ([119], v. III, pp. 1-37), of which we should not speak here. But it must be noted that in the later development of this theory,$^{10}$ Frobenius always remains aware of its relation to the theory of algebras (on which Dedekind had besides never ceased insisting in his letters); and after having introduced for groups the notion of irreducible representation and of completely reducible representation ([119], v. III, pp. 82-103), and shown that the regular representation contains all the irreducible representations, it is by analogous methods that he proposed, in 1903, to take up again the theory of Molien-Cartan ([119], v. III, pp. 284-329). With Burnside [44 a] and I. Schur [279 c], the "hypercomplex" aspect of the theory does not occur explicitly; but it is with them that the fundamental properties of irreducible representations, Schur's lemma and Burnside's theorem come to light. Finally, it must be noted for our objective that it is in this theory that appear for the first time two particular cases of the commutation theorem: in the thesis of I. Schur [279 a] who links up (precisely by commutation in the ring of endomorphisms of a tensor space) the representations of the linear group and those of the symmetric group, and in his work of 1905 [279 c], where he shows that matrices permutable with

$^9$ Let us take note that in the "Grundlagen der Geometrie", Hilbert had given an example of a non commutative field of infinite rank over its centre ([163 c], pp. 107-109). $^{10}$ Part of the results of Frobenius had been obtained independently by T. Molien in 1897 [224 b].

all the matrices of an irreducible representation over the field $\mathbf{C}$ are scalar multiples of $I$ (a result that flows out also from Burnside’s theorem).

It remained to disentangle clearly the common substrate of these theories: it was the work of the German school around E. Noether and E. Artin, in the period 1921-1933 which sees the creation of modern algebra. Already, in 1903, in a memoir on the algebraic integration of linear differential equations ([251 a], v. III, pp. 140-149), H. Poincaré had defined, in an algebra, left ideals and right ideals and the notion of minimal ideal; he had also remarked that in a semi-simple algebra, every left ideal is the direct sum of its intersections with the simple components, and that in the algebra of matrices of order $n$, the minimal ideals are of dimension $n$; but his work remained unnoticed by algebraists.$^{11}$ In 1907, Wedderburn defines anew left and right ideals of an algebra and proves some properties (notably that the radical is the largest nilpotent left ideal ([328 b], pp. 113-114)). But we must wait until 1927 for these notions to be used in an essential way in the theory of algebras.$^{12}$ Putting into general form procedures of proof appearing previously here and there,$^{13}$ W. Krull in 1925 [187 a] and E. Noether in 1926 [236 b] introduce and use systematically the maximal and minimal conditions; the first is used to extend to abelian groups with operators (which he defines on this occasion) Remak’s theorem on the decomposition of a finite group into a direct product of indecomposable groups, whereas the second brings these conditions into the characterisation of Dedekind rings. In 1927, E. Artin [7 c], applying the same idea to non commutative rings, shows how, by a systematic study of minimal ideals, Wedderburn’s theorems can be extended to all rings whose left ideals satisfy both the maximal and minimal conditions.$^{14}$

On the other hand, Krull, in 1926 [187 b], makes the link between the notion of abelian groups with operators and that of linear representations of groups; a point of view generalised to algebras and developed in detail by E. Noether in a fundamental work of 1929 [236 c] which, by the importance of the ideas introduced and the clarity of the exposition, deserves to stand

$^{11}$ Let us also note that, in this memoir, Poincaré observes that the set of operators, in a group algebra, that annihilate a vector of a linear representation space of the group, form a left ideal; he points out that this remark could be applied to the theory of linear representations ([251 a], v. III, p. 149), but never developed this idea.

$^{12}$ It is interesting to remark that, in the interval, the notion of left or right ideal appears, not in the study of algebras, but in a work of E. Noether and W. Schmeidler [238], devoted to rings of differential operators.

$^{13}$ The maximal condition (in the form of “ascending chain condition”) goes back to Dedekind who introduced it explicitly ([79], v. III, p. 90) in the study of ideals of an algebraic number field; one of the first examples of an argument using a “descending chain” is no doubt that which is found in the memoir of Wedderburn of 1907 ([328 b], p. 90) concerning twosided ideals.

$^{14}$ In 1929, E. Noether showed that for rings without radical, these theorems apply assuming only that the minimal condition holds ([236 c], p. 663); C. Hopkins proved in 1939 that this condition alone forces the radical to be nilpotent [167].

alongside the memoir of Steinitz on commutative fields as one of the pillars of modern linear algebra.$^{15}$

Finally, in a series of works which start in 1927 ([237], [34 a], [236 d]), E. Noether and R. Brauer (who is joined starting in 1929-31 by A. Albert and H. Hasse) take up again the study of left fields at the point where Wedderburn and Dickson had left it. If the most important part of their results consists in a deep study of Brauer groups (in particular over the field of algebraic numbers) and so goes beyond the framework of this note, let us point out in any case that it is in the course of this work that the commutation theorems are stated precisely, as well as the notion of a field neutralising a simple algebra and its relations with the maximal commutative subfields; finally, in 1927, Skolem characterises the automorphisms of simple rings [286 b], a theorem found again a few years later by E. Noether [236 b] and R. Brauer [34 a].

So, in 1934, the elementary theory of simple and semi-simple rings has just about reached its final aspect (for an exposé of the state of the theory at this stage, see [87]); since then, it has been developing in two different directions, that we limit ourselves to mentioning briefly. On the one hand, the theory of "factor sets" of R. Brauer and E. Noether has recently received new impulsion, following its incorporation into modern Homological Algebra.$^{16}$ On the other hand, there have been many attempts, with more or less success, to extend — at least in part — the results of the classical theory to rings without minimal condition$^{17}$ or to rings without identity. But to date these extensions have had hardly any repercussions on other branches of mathematics; for more details on this work, we refer you to the recent exposé of N. Jacobson [172 b].

$^{15}$ It is there that is found among other things for the first time in their general form the notions of homomorphism of a group with operators, of the opposite ring, of bimodule, as well as the famous "isomorphism theorems" (which already appear for commutative groups in [236 b]). Particular cases or corollaries of these latter had, it is well understood, come up long before, for example (for the second isomorphism theorem) with Hölder in connection with finite groups [165], with Dedekind concerning abelian groups ([79], v. III, pp. 76-77), with Wedderburn concerning twosided ideals ([328 b], pp. 82-83); as for the first isomorphism theorem, it is for example stated explicitly by de Ségurier in 1904 ([86], p. 65).

$^{16}$ We should not do here the history of this theory and its relations with the notion of the extension of one group by another; but it is appropriate to note that the first "factor sets" made their appearance precisely in connection with a problem of group extensions, in the memoir of 1904 where I. Schur founded the theory of "projective representations" of groups [279 b].

$^{17}$ From 1928, Krull had extended to arbitrary semi-simple modules the general theorems on semi-simple modules of finite length ([187 c], pp. 63-66).