Tagged Questions
4
votes
1answer
69 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
67 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 ...
1
vote
2answers
176 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 ...
5
votes
2answers
230 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 ...
8
votes
3answers
378 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 ...
10
votes
3answers
347 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)?
