On the Cofiniteness of Certain Local Cohomology Modules for a Pair of Ideals

 Let  be non-negative integers and  be an -module such that  is finitely generated for all  and  for all , where  is a class of modules. We hence prove that (1) if  then  is -cofinite for all ; (2) if  and  is finitely generated for all , then  is -cofinite for all . These extend the results of Khazaei-Sazeedeh [10, Thm 2.10, Thm 2.11] for local cohomology modules for a pair of ideals. Finally, we prove that  is -cofinite for all  whenever  is principal,  is an -module satisfying  is finitely generated for all , and  is in dimension . This extends a theorem [13, Thm 1] of Kawasaki.