Divisibility of Ideals

Question

I'm an undergraduate student, who's in my final semester in university.

I have a research project, but the advisor isn't the best.

He said why won't we develop the notion of divisibility of ideals in a ring, and he defined it as follows: If $I, J$ are ideals in a commutative ring $R$, then we say that $$I \mid J \iff \exists K \subset R\ (I = JK)$$ where $K$ is an ideal of $R$.

I was like why don't we define it similarly to our definition of the divisibility in integers, because it will inherit so many properties, that: $$I \mid J \iff \exists K \subset R\ (J=KI) $$ where $K$ is an ideal of $R$.

He said because I in his definition is smaller than J, and that I can't know because I'm just an undergrad student, and that he has a PhD so I shouldn't question him, and I should just work on what he gave me, so I did for a while, and when it didn't make any sense to me, I stopped and returned to working on my own definition -which then I found out it's not mine- and he disappeared for two weeks, and I couldn't reach him.

Nonetheless I kept looking and reading in a number of graduate books looking for ideas to study the properties of this definition, I written 7 papers and when I was really excited to show him what I came up with.

When we had our meeting, I told him that his definition didn't make sense to me so I worked on mine and when I said that he threw the notebook away, refused to even see what I had written, and told me that I don't have the mathematical maturity to even discuss this with him, and just the fact that I disobeyed him shows that I'm not a serious person.

I was shocked by what he said, I couldn't even reply to that, I'm a very emotional person, and his words really hurt me.

So I kept looking for ways to prove my point to him, and found a paper by Keith Conrad, that uses exactly the definition I used, but I want to come up with more references that does this, or even if any that use his definition, just to know who is right.

I can't find any book that does this, can anyone help me with this?

Answer 1

He said because $I$ in his definition is smaller than $J$, and that I can't know because I'm just an undergrad student, and that he has a PhD so I shouldn't question him

What is the subject area of his PhD? It can't be an area that uses divisibility of ideals, because what he says is absurd. Even in the integers, as you noticed, divisibility corresponds to reverse containment of ideals: $a \mid b$ in $\mathbf Z$ if and only if $a\mathbf Z \supset b\mathbf Z$. Consider that the prime numbers in $\mathbf Z$, which are the "smallest" elements in terms of divisibility, generate the maximal ideals. The fact that a positive integer $a$ has only finitely many factors corresponds to $a\mathbf Z$ being contained in finitely many ideals of $\mathbf Z$. The ideal $a\mathbf Z$ contains infinitely many ideals since $a$ has infinitely many multiples. This is why the role of $\subset$ and $\supset$ for ideals behaves in a way opposite to that suggested by the numerical relations $<$ and $>$.

The definition of divisibility of ideals that you read in my notes is used by everyone except your advisor. It can be found in basically every textbook on algebraic number theory. Some textbooks may not actually use the divisibility notation, so I'll limit myself below to several books using it that I found on my shelf. Before I give the list, note that according to your advisor's suggested notation $I \mid J$ implies $J \supset I$, while according to your notation, $I \mid J$ implies $I \supset J$.

1. Lang, Algebraic Number Theory p. 20: for (fractional) ideals $\mathfrak a$ and $\mathfrak b$, "we say $\mathfrak a \mid \mathfrak b$ if and only if there is an ideal $\mathfrak c$ such that $\mathfrak a \mathfrak c = \mathfrak b$".

2. Alaca and Williams, Introductory Algebraic Number Theory, p. 207: for (fractional) ideals $A$ and $B$, "$A \mid B$ if there is an integral ideal $C$ such that $B = AC$".

3. Marcus, Number Fields, 1st edition, p. 59: for ideals $A$ and $B$ in a Dedekind domain $R$, "$A \mid B$ iff $A \supset B$". The first step of that proof says "$A \mid B \Rightarrow A \supset B$".

4. Stewart and Tall, Algebraic Number Theory, 2nd edition, p. 121: "$\mathfrak a \mid \mathfrak b \Longleftrightarrow \mathfrak a \supset \mathfrak b$".

If you ask your advisor to show you some textbooks on algebraic number theory that use his definition of divisibility of ideals then he won't be able to find any, unless he writes one himself.

Answer 2

If $a,b$ are integers, then $a\mid b$ iff there is an integer $c$ such that $b=ac$.

The natural generalization to ideals is the following: $I\mid J$ iff there is an ideal $K$ such that $J=IK$.

Reference.
S. Lang, Algebra, 2002; page 116, exercise 17.
D. Dummit, R. Foote, Abstract Algebra, 2004; page 767.