## “School of Mathematics”

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Paper   IPM / M / 92
School of Mathematics
Title:   Asymptotic prime ideals related to exact functors
Author(s):
 1 K. Divaani-Aazar 2 M. Tousi
Status:   Published
Journal: Comm. Algebra
No.:  8
Vol.:  27
Year:  1999
Pages:   3949-3968
Supported by:  IPM
Abstract:
Let I be an ideal of the commutative ring R and let \scrptCR denote the category of R-modules and \scrptCN (resp. \scrptCA) be the subcategory of Noetherian (resp. Artinian) R-modules. Let N denote a Noetherian R-module and N′ be a submodule of N. For a linear exact covariant (resp. contravriant) functor T:\scrptCN→ \scrptCR, AssR(T(N)) (resp. AttR(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 AssR([(T(N))/(InT(N′))]) and AssR([(InT(N))/(InT(N′))]) (resp. AttR(T(N/N′):T(N)In) and AttR(T(N/N′):T(N)In/0:T(N)In)) are eventually constant for large n. Also, the dual results are shown to be true for a linear exact functor T:\scrptC A→ \scrptCR.