# History of Direct Sums and Direct Products

So I like to get down into the details of how certain mathematical concepts came to be, and purely as a matter of curiosity, I was wondering if anyone know which mathematician first gave the definition of a direct sum?

I’ve always just sort of assumed that since for a finite set of modules, the direct sum and the direct product is the same, people probably came up with the direct product first, and then later, some mathematician wanted to do something with infinite sets of modules, realized that the universal property (of direct sums) doesn’t necessarily hold in that case, and so invented the direct sum to ”get around this,” but I’ve been wondering whether we actually know that this is what happened.

Any takers?

• Direct sums and products (and homorphisms and quotients and…) all existed before the concept of universal mapping properties.
– KCd
Nov 21 at 0:38

"Direct sum" and "direct product" did not use to mean what they now mean in the OP sense, and even today the old usage persists. Van der Waerden in Moderne Algebra (1930-31) (the following section numbers are given according to the 1967 edition) uses "direct sum" when additive convention is assumed for the operation (e.g. for rings and modules, §92) and "direct product" when multiplicative convention is (e.g. for general groups, §53). He also uses "direct product" for what is now called tensor product (§67) in a special context of Galois theory, but that section was added in 1967. Van der Waerden's book is based on Artin's and Emmy Noether's lectures from mid 1920s. Sum/product distinction based on the additive/multiplicative convention is still current in group theory.

Murray and von Neumann used "direct product" for tensor product (of vector spaces) from 1936 on, they also pioneered the symbol ⊗, see History: Direct Product became Tensor Product? This is related to similar use for matrices, e.g. in MacDuffee, The Theory of Matrices (1933), where "direct sum" stood for combining matrices block-diagonally, and "direct product" for 'Kronecker' product. Kronecker had little to do with it, but the name stuck after Hensel c. 1890 (Zehfuss introduced it back in 1858, see Henderson et al., On the history of the Kronecker product). "Direct product" is still occasionally used.

Bourbaki decided to tidy it up and sort it out in their Algebra I (multilinear algebra, 1948). There they reserved "direct product" (of modules) for the Cartesian product (regardless of conventions about the operation), "direct sum" for its subset with only finitely many non-zero entries, and separated it from "tensor product" following Whitney's 1938 general definition, see Origin of the modern definition of the tensor product. All of this happened before categorical notions of universal property, product and coproduct became fashionable in 1950s. But yes, Bourbaki's cleanup was likely motivated by working with infinite sums and products.

• Excellent! Just what I wanted! Thanks! :) Nov 21 at 7:11
• Also, I was able to eventually track down what may very well be the first use of the term "direct product" in group theory. In an article by G. A. Miller from 1899, he uses the term "direct product" to mean what we would use "direct sum" to mean (i.e. the sense in which van der Waerden later uses it), and he says in that article that the first use of it occurs in an article by Otto Hölder from 1893. I checked up the Hölder article, and there he says, "Ich nenne in diesem Fall die gebildete Gruppe das directe Product..." which makes it clear that this is his term. So there's that genesis. Nov 24 at 11:16
• @Conifold which edition of Moderne Algebra were you looking at when mentioning section numbers? You say direct products of general groups are in Section 53, but the 1930 edition of Volume 1 has direct products of groups in Section 42 (and Section 53 is about Galois theory of polynomials of degree 2, 3, and 4). Section 67 is about the algebraic theory of real fields and doesn't have anything resembling tensor products in it. What is the title of the section you saw where tensor products (as "direct products") are used in Galois theory?
– KCd
Nov 25 at 1:05
• @KCd It was 1967 edition, the title was Normal Bases. Nov 25 at 1:13
• Oh, then please mention that in your answer! Since the OP's question is about the origin of terminology, and you refer to "Moderne Algebra (1930)", I thought your section numbers were from the original 1930 edition. An edition from the 1960s isn't an ideal resource since the content of that book changed in new editions. In particular, the normal basis theorem as a context for tensor products (even when called direct products) isn't useful since that theorem was not proved for infinite fields until 1932, which is after the 1st edition of Moderne Algebra came out (1930, 1931).
– KCd
Nov 25 at 1:35