The finiteness of certain sets of attached prime ideals and the length of generalized fractions

We introduce the notion of strict f-sequence and apply this concept to study the finiteness of asymptotic sets of attached prime ideals of local cohomology modules of M, to study the polynomial property of the length of generalized fractions as defined by Sharp and Hamieh (J. Pure Appl. Algebra 38 (1985) 323), and to characterize pseudo-generalized Cohen–Macaulay modules defined in Cuong and Nhan (J. Algebra 267 (2003) 156).