Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I'm trying to determine all the prime filters in the lattice $N = (\mathbb{N}, \text{lcm}, \text{gcd})$, where the order is given by divisibility.

I know that this lattice is distributive and that its join-irreducible elements are precisely the positive powers of any prime number. I have proved that for any join-irreducible element $x \in N$ it is the case that $N \backslash \uparrow x$ is a prime ideal. Since $\uparrow x$ is a prime filter, I'm wondering if every prime filter in $N$ has this form. Any ideas?

share|improve this question
add comment

1 Answer 1

up vote 1 down vote accepted

$\newcommand{\lcm}{\operatorname{lcm}}$Suppose that $F$ is a (proper) prime filter in $N$. Let $m\in F$ be minimal. If $m$ is not a prime power, it has a non-trivial factorization $m=ab$ such that $\gcd(a,b)=1$. Then $m=\lcm(a,b)$, so $a\in F$ or $b\in F$, contradicting the minimality of $m$. Thus, every minimal element of $F$ is a prime power, and it follows immediately that $F={\uparrow m}$.

share|improve this answer
    
That's a cute argument, Brian. Thanks! –  ragrigg Nov 7 '12 at 6:41
    
@ragrigg: You’re welcome! –  Brian M. Scott Nov 7 '12 at 8:00
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.