# Galois connection for annhilators

Mathematics Asked on November 30, 2020

https://en.m.wikipedia.org/wiki/Annihilator_(ring_theory)
Under the section "Category-theoretic description for commutative rings"
Wikipedia states few results for galois connection for annihilator for modules.
$$1. ann(ann(ann(S)))=S$$ and

$$2.span(S)subset ann(ann(S))$$.

It makes me wonder if such results exists for annhilators of ideals in the ring.

I tried few easy examples

$$span(S)subset ann(ann(S))$$ this holds in any integral domain. $$ann(ann(ann(S)))=S$$ this too holds.

I wonder if these two results is true in general. ( I doubt 1 maynot be hold)

Also if $$span(S)subset ann(ann(S))$$ holds in general, under what condition they are equal.

One condition I found is if $$span(s)$$ is generated by an idempotent then equality holds but I can’t think anymore general.

I think you must have typoed the first one. I think you rather mean

$$ann(ann(ann(S))) = ann(S)$$

because, for example, if you take the zero ideal in $$mathbb Z$$, your first equation is wrong as written.

I'm going to use some ad hoc notation to write the general case.

Let $$M$$ be a right $$R$$ module and write

$$ann(X)={rin Rmid Xr={0}}$$ for a nonempty subset $$Xsubseteq M$$

$$Ann(Y)={min Mmid mY={0}}$$ for a nonempty subset $$Ysubseteq R$$

Now note:

1. Just by the definitions of annihilators, $$Ysubseteq ann(Ann(Y))$$ and $$Xsubseteq Ann(ann(X))$$.

2. Furthermore, it is easy to check that both maps $$ann()$$ and $$Ann$$ are containment reversing.

Now, using the first equation, substituting $$ann(X)$$ for $$Y$$, you'd get $$ann(X)subseteq ann(Ann(ann(X)))$$. On the other hand, applying $$ann$$ to both sides of $$Xsubseteq Ann(ann(X))$$, you get $$ann(X)supseteq ann(Ann(ann(X))$$. Therefore $$ann(X)=ann(Ann(ann(X)))$$.

Using a similar argument, you have $$Ann(Y)=Ann(ann(Ann(Y)))$$.

From this, you can derive that the left and right annihilator maps in $$R$$ satisfy $$ell r ell = ell$$ and $$r ell r=r$$, and in the case of a commutative ring you'd have the identity you proposed above: $$ann^3=ann$$.

As for the second part, from $$Ysubseteq ann(Ann(Y))$$ you have that the left ideal generated by $$Y$$ is contained in $$ann(Ann(Y))$$ since $$ann(Ann(Y))$$ is a left ideal of $$R$$.

And likewise, you have that the submodule generated by $$X$$ is contained in $$Ann(ann(X))$$ since $$Ann(ann(X))$$ is a submodule of $$M$$.

But in general it does not have to be the case that you have equality.

You can work out why $$ann(Ann(L))=L$$ if and only if $$L$$ is of the form $$ann(N)$$ for some submodule $$N$$ of $$M$$.

Likewise $$Ann(ann(N))=N$$ if and only if $$N=Ann(L)$$ for some left ideal $$L$$ of $$R$$.

When $$M$$ is the ring itself, ideals like these are called "annihilator ideals" or sometimes "annulets" but I am not completely sure what they call them in the general case when looking at the annihilator maps between a module and its ring.

Correct answer by rschwieb on November 30, 2020