Specifically: Invertability of all elements with respect to the multiplication leads to the notion of division algebra and these have been studied for a long time. (https://en.wikipedia.org/wiki/Division_algebra)
When studying math at german universities, alebras are something you'll encounter in your 2nd year (latest; but might already show up in 1st year analysis albeit with a different focus). Implicitly, division algebras show up a lot when students learn about field extensions, galois theory and the algebraic closure of a field (usually 3rd semester). A more general treatment of division algebras is not a common subject, though.
The basic object that the author seems to be interested in is that of an "algebra over a field" (https://en.wikipedia.org/wiki/Algebra_over_a_field).
Specifically: Invertability of all elements with respect to the multiplication leads to the notion of division algebra and these have been studied for a long time. (https://en.wikipedia.org/wiki/Division_algebra)
When studying math at german universities, alebras are something you'll encounter in your 2nd year (latest; but might already show up in 1st year analysis albeit with a different focus). Implicitly, division algebras show up a lot when students learn about field extensions, galois theory and the algebraic closure of a field (usually 3rd semester). A more general treatment of division algebras is not a common subject, though.