History of Direct Sums and Direct Products

Question

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?

Answer 1

"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.

Answer 2

Basically, with the invention of sets and modern algebra it was discovered that one can not only add, subtract, multiply and divide things of an algebraic gadget like a group or a ring. But you can also do these things at the level of an algebraic gadget. For example, you can multiply two groups together or add two rings together.

However, whilst there was a profusion of new ways of adding and multiplying and there was a family resemblence between these constructions there were enough differences that a systematic method of saying what it means to add two objects together and how to multiply two things together in a systematic fashion was lacking.

Well, this was the case until category theory appeared. Then it was understood that addition, called here the coproduct, was dual to multiplication, called here the product. And this was systematic in all categories. Moreover, the product was an example of a limit and the coproduct, an example of a colimit.

It turns out that the direct sum and direct product for vector spaces are isomorphic (ie the same) for finite-dimensional vector spaces but differ for infinite-dimensional spaces. Hence, their differences, at tge finite-dimensional level is not apparent. In infinite dimensions, it is the direct sum that is the coproduct whilst the direct product is the product. Its probably worth adding that the tensor product of algebras, a vector space equipped with a notion of multiplication, is the coproduct in the category of algebras which shows that coproducts can be quite different to what one might imagine them to be.

Since Peano gave the first formal description of vectors in 1888, the notion of a direct sum and product of vector spaces will have to be after this. Its worth noting that Peano axioms were inspired by Grassmann's invention of the exterior algebra, his Die Lineale Ausdehnungslehre first published in 1844.