What is the significance of $A+ (B\cap C)=(A + B)\cap C$, where $A\subseteq C$, for modules?

Question

My book (Introduction to Ring Theory, Paul Cohn) states this as a theorem and gives a proof. The book usually skips over trivial/easy proofs, so I don't really understand why this is in here.

Isn't the statement absolutely obvious for sets $A,B,C$ with $A\subseteq C$? You just need to draw a diagram and it becomes immediately obvious!

Is there a reason why it would not be obvious in modules?

Answer 1

I'm not sure if the statement is obvious to you but in general $+$ and $\cap$ don't distribute over each other. I wouldn't expect them to either since $\cap$ is a set theoretic operation and $+$ is only defined for modules/ideals/abelian groups. However, when $A\subset C$ they do distribute over one another.

But the proof shouldn't go something like this:

$$A+(B\cap C)=(A+B)\cap (A+C)=(A+B)\cap C $$ where the last equality comes from $A\subset C$ because here we assumed they distribute in the first place which is false in general. So any proof of this fact can't use the above method (or the other direction).

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An explicit example that these operations don't distribute is by taking the submodules of the $\mathbb{Z}[X]$-module $\mathbb{Z}[X]$ where we can now just look at ideals

$A=(2)$, $B=(x-1)$, and $C=(x+1)$ then $$A+(B\cap C)=(2)+(x^2-1)=(2,x^2-1)$$ and $$A+ B \cap A+ C=(2,x-1)\cap(2,x+1)=(2,x+1)$$

since $(2,x-1)=(2,x+1)$.

Answer 2

Judging from your diagram, it looks like (?) you are interpreting $+$ as set union. Certainly, if you are working with sets using union and intersection, your picture is accurate.

But using using the $+$ to join submodules of a module does not behave the same way. If $A$ and $B$ are submodules of a module, $A+B$ contains many more elements than just $A\cup B$ in general.

To summarize the situation, the property proved is that the lattice of submodules of a module (join $+$ and meet $\cap$) is a modular lattice. The lattice of subsets of a set, on the other hand (join $\cup$, meet $\cap$), is also modular, but even more strongly it is a distributive lattice. In general, the lattice of submodules of a module does not have to be distributive.

For more about modularity, let me refer you to a question we had some time ago on the topic: Why are modular lattices important?