Yes, I was misremembering. My mistake!

I very strongly disagree. Without considering the whole ring to be an ideal, you can't even define "the ideal generated by some elements" because there may not be such an ideal, since it could be the whole ring. Likewise, you can't perform common operations on ideals like sums because they could result in the whole ring. In fact, I would say there is almost no advantage in excluding the ring itself from the set of ideals.

Just to check, I have three math textbooks from my college days that include the definition of an ideal, and none of them attempt to exclude the ring itself from the definition.

The obvious compromise is to introduce the concept of a proper ideal as an ideal that is a proper subset, and to use that when you need to exclude the ring itself. E.g., a maximal ideal is a proper ideal that is maximal with respect to inclusion.

That's not my experience at all as a mathematician. "Maximal ideal" implicitly means "maximal proper ideal", yes, but generally ideals don't have to be proper unless specified so.

If you don't include the whole ring as an ideal, you can't even define ideal addition, etc. I took an algebra class once from a professor who decided to define "ideal" to mean "proper ideal". After a few weeks he had to give it up because it just became too much trouble for reasons like that; he had to too often say "possibly improper ideal", i.e., this convention had the opposite effect he intended! I can't think of any other source I've seen use that convention.

Historically, mathematicians have spent a huge amount of time and effort formulating optimal axioms and foundations so that theorems would follow naturally from structure. Theorems following “trivially” from a theoretical framework that took years to develop isn’t an indictment of the theorem, but an endorsement of the incredible effort expended to develop an optimal context for expressing and understanding the theorem.

> Trivial in math is a term that refers to anything you've already learned.

According to a professor, "trivial" means: "If this is not trivial for you, you should see this as a clear signal that you should take this course seriously instead of slacking of, or even that you simply are in the wrong course."

My pet peeve math term is "clear". A long time ago I thought could teach myself group theory by buying the Springer group theory book and reading it from chapter 1, 1 page at a time. But I was blocked within the first 5 pages because the axioms and first few proofs kept saying how "clear" it was that all the results followed. Unfortunately, it was not "clear" to me :(

Indeed. Famously, though, figuring out if something is a tautology is undecidable!

ironically the article itself uses "trivial" in its other, more mathematical sense :)

this is a standard thing in "mature" areas of math and it's absolutely the opposite of what's good for the student (all of the machinery being hidden in the definition instead of developed in the theorem's proof).

EDIT: if you hate "a monad is a monoid in the category of endofunctors" then you also hate "definitions should be hard and theorems easy".

> How to distinguish mastery of a complex subject from parsing one formally expressed in a complex way?

In my opinion: the difference between a complex subject and one formally expressed in a complex way is that in the former, the results that you get are really deep (understanding them at the end feels like a spiritual experience).

I think what you call “parsing” is largely indistinguishable from mastery in a lot of fields, particularly in abstract mathematics

It depends a bit on your tastes. The content falls under the broad umbrella of Abstract Algebra, more specifically Ring Theory, or perhaps Field Theory if you squint a bit. Those are your keywords.

"Applications of Abstract Algebra with Maple and MATLAB" by Klinger, Sigmon, and Stitzinger is apparently good for those with an engineering background: https://www.maplesoft.com/books/details.aspx?id=624.

If you're committed, then any introductory text on abstract algebra or group theory might capture your interest.

I would recommend starting with applications or something as close to your wheelhouse as possible just to stay motivated. Abstract algebra, in particular, is known for requiring quite a lot of machinery before obviously connecting with other things, which can feel like an onslaught unless you're inherently interested.

Have fun though! It's really one of the deepest subjects in modern math, IMHO. Almost every field has been affected by it's results.