If I wasn't familiar with that concept already, then I would probably assume that math is no more rigorous than psychology after encountering this thread.
The exact disciplines are doing themselves a terrible disservice by muddying up established terminology like this (and "algebraic integers" are far from the only such case).
Mathematicians pretty reliably say "algebraic integer" or "integer in [some specific class of numbers that has non-rational integers in it]" when they are talking about the broader notion, and if they're doing something where that broader notion is often relevant they will generally say something like "rational integer" when they mean the narrower notion. So in practice there is seldom any confusion.
And algebraic integers really _are_ like ordinary integers in important ways. Inventing a completely new term would not obviously be an improvement.
It's not like this sort of thing is unique to mathematics. Once upon a time a "language" was a thing human beings used to communicate with one another. Then along came "programming languages" which are not languages in that sense. And then things like "hypertext markup language" which isn't a language in the programming sense either.
(Arguably this is partly mathematicians' fault since I think they were the first to use "language" to refer to purely formal constructs. But I think the use of "language" in computing arose mostly by analogy to human languages.)
And it happens plenty outside "the exact disciplines". A republican is someone who favours a mode of government that doesn't have monarchs, but if you call someone a "Republican" in the US you mean something rather more specific and
a few "Republicans" would actually quite like a system hard to distinguish from monarchy. A window is a transparent thing placed in a wall to let light in, but a window of opportunity is something quite different. A czar is the absolute ruler of Russia, but when someone says (rightly or wrongly) that Kamala Harris was "border czar" they don't mean that. A star is a gigantic ball of stuff undergoing nuclear fusion and producing unimaginable amounts of energy, but even the most impressive rock stars don't do that, and some people called "rock stars" have never played or sung a note of rock music in their lives.
The red herring principle[1] is unfortunately popular enough in mathematical terminology to have a name and a page about it.
Roughly, a fooish bar will frequently be something like a bar except fooish, so not actually a bar. (Algebraic integer, multivalued function, manifold with boundary, etc.) On the other hand, a nonfooish baz when baz is normally fooish often means a not necessarily fooish baz, so a particular one might be fooish but we can’t assume that. (Noncommutative ring, nonassociative algebra, the very field of noncommutative geometry, etc.)
> The exact disciplines are doing themselves a terrible disservice by muddying up established terminology like this
So, what term do propose they use for this? “Hutyfreklop” or “gensym_167336871904” would probably be unique, but wouldn’t tell anything about the subject itself, “roots of polynomials with integer coefficients and a leading coefficient of one” would get cumbersome soon.
Terminology is often times used to encapsulate a lot of information in a single word or phrase. It’s sort of a compression of information to facilitate communication. Things like “roots of f” is a shorter way to say that: “the set of all x such that f(x) = 0”. As you get deeper into a subject the more terminology you encounter. This is why research papers are generally unintelligible to those with no training in the areas that the research is about. To not use terminology would make papers insanely long and far too tedious to read.
Mathematicians (with books in hand) tend to have the ability to disambiguate when confusion arises. But they rarely do, instead relying on context and the intelligence of the reader to derive meaning.
The exact disciplines are doing themselves a terrible disservice by muddying up established terminology like this (and "algebraic integers" are far from the only such case).