## Unsolved problems concerning Artinian Rings and Artinian Modules

I am preparing a write-up on Topics on Artinian Rings and Modules for a project. I hope to mention some unsolved problems in the domain of these objects along the way. Till now, I have been able to find and understand the statement of the Monomial Conjecture. Upon googling, I did not find problems that would serve my need as they were well over my head.

Question: Are there more well-known and simply-stated problems?

P.S.: By simply-stated, I mean understandable by a student comfortable with some theory of modules and rings, chain conditions on rings and modules, radicals(nil and Jacobson's), Nakayama's lemma and the Krull dimension of a ring plus some special topics. You can assume this to be my 'bailiwick' while suggesting problems.

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There are many open problems which are fairly easy to state, also one might need some basic definitions, for example derived functors. I will provide mostly pointers to the ones I know, you can google for more details (one can easily fill many projects with each topic below). Interestingly, most people do not consider the Monomial conjecture as a problem about Artinian rings (the word Artinian only appear as in the definition of a system of parameters).

Representation theory of Artin algebras: you can start with the list at page 411 of this book.

Numerics of Betti numbers, Hilbert functions: perhaps the most famous one is the Buchsbaum-Eisenbud-Horrocks Conjecture;

If $R$ is a regular local ring of dimension $n$ and $M$ is an Artinian $R$-module, then the rank of the $i$-th module in the minimal free resolution of $M$ is at least $n \choose i$. (weaker version, but as open: the sum of the ranks is at least $2^n$, graded version is also open).

An important open problem is to characterize the Hilbert functions of Gorenstein graded artinian algebras. See the surveys by Irena Peeva or Valla for some other questions.

Homological problems: Here the big one, and also one of the simplest looking, is the Auslander-Reiten Conjecture

If $R$ is commutative Artinian ring and $M$ is a finitely generated $R$-module such that $\text{Ext}^i(M, M\oplus R) = 0$ for all $i>0$, then $M$ is projective.

Mathoverflow: see for example here or here.

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 Dear HD, thank you very much. I really need to know the "new" terms (to me) as I did not understand most of them on first reading. However, I never thought such a list would exist. Much obliged. +100 – To be cont'd Apr 9 2011 at 12:10