# Tagged Questions

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### On modules over simple rings

We suppose that all rings are left Artinian simple rings and all modules over a ring are of finite length. Let $M \neq 0$ be a left module over a ring $R$. By Wedderburn theorem, $R$ is a matrix ring ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...