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Paper IPM / M / 92  


Abstract:  
Let I be an ideal of the commutative ring R and let
\scrptC_{R} denote the category of Rmodules and
\scrptC_{N} (resp. \scrptC_{A}) be the
subcategory of Noetherian (resp. Artinian) Rmodules. Let N
denote a Noetherian Rmodule and N′ be a submodule of N. For
a linear exact covariant (resp. contravriant) functor
T:\scrptC_{N}→ \scrptC_{R},
Ass_{R}(T(N)) (resp. Att_{R}(T(N))) is determined
and as a consequence several results concerning asymptotic prime
ideals are deduced. For example, it is shown that both sequences
of sets Ass_{R}([(T(N))/(I^{n}T(N′))]) and
Ass_{R}([(I^{n}T(N))/(I^{n}T(N′))]) (resp.
Att_{R}(T(N/N′):_{T(N)}I^{n}) and
Att_{R}(T(N/N′):_{T(N)}I^{n}/0:_{T(N)}I^{n})) are eventually
constant for large n. Also, the dual results are shown to be
true for a linear exact functor T:\scrptC_{
A}→ \scrptC_{R}.
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