Due to whom is the modern (i.e. via its universal property) definition of the tensor product, and in which article was it communicated?
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My recollection agrees with Mariano. Here's an except from Weibel's History of homological algebra
and from p. 172 of Mac Lane's Homology
and from Whitney's Collected Papers v. 1
is usually credited as the original source.
Very soon after that, Bourbaki crystallized the modern definition.