We know the structure of f.g. torsion modules over a PID. What can we say if ring is not a PID, like only local. Any result about structure when the ring is not a PID..

Big list? Community wiki? Fishing?
– Yemon ChoiNov 19 '10 at 18:25

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Presumably we are dealing with finitely generated modules? Then you can make statements about the support / the associated primes and even various primary ideals... You aren't going to have pretty statements (no direct sums), but much can be still said. Have you looked at on commutative algebra, especially sections on primary decomposition. For example, see Chapter IV, section 2 of Bourbaki's commutative algebra. My response, is probably not an appropriate answer for a mathoverflow question (ie. my reading of the question indicates that the question is not mathoverflow appropriate)
– Karl SchwedeNov 19 '10 at 20:29