The Two Ideals of Fields
52 comments
·May 31, 2025kevinventullo
mathgradthrow
Those sources are crazy. The ideals form a lattice under intersection and "+".
All of algebraic geometry (a very popular branch of mathematics for studying rings) is built on the lattice of ideals. There is no way of giving a ring this structure without the ring itself being the top element of this lattice.
What you probably mean to say is that there are sources that do not treat R as a prime ideal of itself.
kevinventullo
Yes, I was misremembering. My mistake!
mathgradthrow
Ah, I see we really piled on you for this
tome
See also "too simple to be simple": https://ncatlab.org/nlab/show/too+simple+to+be+simple
vouaobrasil
Pretty much most sources, actually. When actually working with ideals, there's almost never an advantage to consider the whole ring an ideal. So all ideals in virtually all the literature I've ever read were taken to be proper, i.e. proper subsets
WCSTombs
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.
kevinventullo
Too late to edit, but you’re right I was misremembering. I was thinking of whether (1) should be considered a prime ideal.
Sniffnoy
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.
vouaobrasil
Another interesting fact about fields is that a commutative ring is a field if and only if every ideal is a prime ideal. (Obviously, every ideal of a field is a prime ideal. The converse is more interesting...)
shiandow
This is one of those cases where the algebraic geometry perspective is actually helpful (without too much trouble).
If the only subspaces are single points then the space is itself a point.
mtsolitary
This is a great example of one of those things in abstract maths that is hard to follow when you learn it, but once you’ve been through it a few times and learnt the definitions to heart it’s really just a rephrasing of the definitions.
tux3
Trivial in math is a term that refers to anything you've already learned.
You sometimes hear people say that math is tautological. But regardless of whether it's all just an elaborate rephrasing of the axioms, it's quite beautiful.
math_dandy
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.
mathgradthrow
Half of the work of mathematics is in correct definitions. Groethendieck referred to the division between mathematical labors as hunting and farming.
This is not my most popular opinion, but probably the most consequential invention of the last 400 years was the set. Suddenly all mathematical knowledge could be verified in one framework. Physicists had a target in which to state their models.
If you could state your hypothesis in the language of mathematics, "everyone" knew exactly what you meant by it, and how to go about testing your claims, or proving them, if they happened to be about mathematics itself.
Calculus was invented in 1690ish, physicists like to claim that this was the most important advance in physics, but quantum mechanics and relativity didn't happen until dedekind invented the real numbers, 200 years later.
It turns out that knowing what you're talking about matters.
aleph_minus_one
> 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."
kevinventullo
I dunno, a common refrain I heard across all fields of math in grad school was “This is obvious. Wait, is this obvious…? Y… yes yeah it’s obvious. ”
tekla
This. It's always a good sign you've fucked up somewhere.
xelxebar
Indeed. Famously, though, figuring out if something is a tautology is undecidable!
zem
ironically the article itself uses "trivial" in its other, more mathematical sense :)
gosub100
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 :(
mwcremer
I had a calc prof who was in the middle of a lecture, "...and as any fool can see, X is..." He stopped, turned around, and said, "You know, sometimes when I say, 'It is intuitively obvious', or, 'As any fool can see', I realize it may not be intuitively obvious, and any fool may not be able to see. But as any fool can see, X is..."
JadeNB
> the Springer group theory book
I am skeptical that this uniquely identifies a book (unless you mean the book "Linear Algebraic Groups" by the author called Springer, rather than the publisher called Springer, in which case it's definitely not the way to start learning group theory!).
srean
Spivak states in his "Calculus on Manifolds' that definitions should be hard (to refine and state) and when done well, the theorems easy.
almostgotcaught
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".
andrewflnr
My first instinct is to agree, but I'm not sure actually. What I really want when learning a new area of math is the full motivation for the tricky definition, taking as much time as needed to follow the dead ends of easier but worse definitions. Then I get the whole picture. IMO the motivation is the key thing for students, not the definition being easy.
Though maybe the way this course would work is in fact by proceeding through a series of easy but explicitly flawed definitions, and proving both real results and nonsense from them, so you see why the real definition is justified.
bubblyworld
Yeah, as an example this is a simple corollary of the following result which gets used a lot: the quotient of a (unital, commutative) ring by a maximal ideal is always a field.
If your ring has only two ideals then the trivial ideal is maximal, and thus your ring is already a field!
The more you know, the more "shortcuts" you start seeing, I guess.
downboots
How to distinguish mastery of a complex subject from parsing one formally expressed in a complex way?
aleph_minus_one
> 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).
smohare
[dead]
mtsolitary
I think what you call “parsing” is largely indistinguishable from mastery in a lot of fields, particularly in abstract mathematics
layer8
Knowing the rules of chess doesn’t make you a chess master. Knowing the syntax and semantics of a programming language doesn’t make you a master software architect.
fn-mote
I disagree. "Parsing" is the first level of understanding. If you are not moving past the parsing level, you have not achieved any kind of mastery.
My experience is that mastery means more like "you have a mental model which gives you 'intuitive' reasons to accurately classify things as true/false and provides some motivation for the reasoning".
An example: you see someone has solved a degree 4 equation by repeatedly applying the quadratic equation, getting 8 solutions. "No way."
Another example: watch a famous baking show and you see somebody put a bunch of different sized pieces of bread in the oven at the same time. Right away: "aren't they going to cook at different rates?" Sure enough, some burned, some raw.
VladVladikoff
Is there a good place to start with this stuff? I’ve done undergrad engineering but this math is alien to me.
xelxebar
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.
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FYI many sources do not count the entire ring as an ideal. If you do, you’d have to define “maximal ideal” to mean “an ideal that is maximal with respect to inclusion, ignoring the entire ring ideal.”