5
votes
1answer
81 views

Is there a ring with the lattice of ideals isomorphic to $(\omega+1)^{\operatorname{op}}?$

In this question, I gave an example of a ring whose lattice of two-sided ideals is order-isomorphic to $\omega+1$. I've been playing a bit with trying to find rings with a given lattice of ideals ...
2
votes
1answer
78 views

Diamonds of ideals, part 3

I'd like to wrap up the line of questioning started first in this question and then continued in this question. The only variant left to try is: "How close can you get to the Diamond lattice ...
11
votes
3answers
664 views

The Chinese remainder theorem and distributive lattices

In The Many Lives of Lattice Theory Gian-Carlo Rota says the following. Necessary and sufficient conditions on a commutative ring are known that insure the validity of the Chinese remainder ...
1
vote
2answers
245 views

Followup to “Examples of rings with ideal lattice isomorphic to $M_3$, $N_5$”

In this post: Examples of rings with ideal lattice isomorphic to $M_3$, $N_5$ a nice example was given of a non-distributive ring. The lattice of ideals turned out to be the Diamond lattice $M_3$ with ...
4
votes
1answer
264 views

Exercise on distributive module lattices

I'm trying to do the very first exercise in Representations and cohomology I by Dave Benson, it's been bugging me for a while now. I don't really know how to start, although I imagine we will need to ...
8
votes
2answers
377 views

Are ideals in rings and lattices related?

There are (at least) two notions of ideals: An ideal in a ring is a set closed under addition and multiplication by arbitrary element. An ideal in a lattice is a set closed under taking smaller ...
11
votes
3answers
570 views

Examples of rings with ideal lattice isomorphic to $M_3$, $N_5$

In thinking about this recent question, I was reading about distributive lattices, and the Wikipedia article includes a very interesting characterization: A lattice is distributive if and only if ...
11
votes
3answers
444 views

Simple example of non-arithmetic ring

Can anyone provide a simple concrete example of a non-arithmetic commutative and unitary ring (i.e., a commutative and unitary ring in which the lattice of ideals is non-distributive)?