Let I, J be ideals of a commutative Noetherian local ring (R,m) andM, be a finite R-module. We prove that, inf{f − depth(a,M)| a 2 ˜W (I, J)} = inf{i|Hi I,J (M) 6= Him (M)} = inf{depthMp| p 2 W(I, J) \ {m}} and inf{f−depth(a,M)| a 2 ˜W (I, J)} is the least integer i such that Hi I,J (M) is not Artinian. By using the concept of the serre class of R–modules, we conclude some properties of local cohomology module with respect to a pair of ideals. In addition, we show that, if M is a finite module over a local ring R, then Hi m,J (M) is not Artinian for some non–zero ideal J of R and some integer i. In the remaining part of the thesis we discuss about finiteness properties of local cohomology with respect to a pair of ideals I, J. Firstly, by extending notion I–cofinite as (I, J)–cofinite, we conclude the main result of dibaei–Yassemi in [13] and [14]. Finally, we prove that, if t = inf{i|Hi I,J (M) 6= 0}, then for all p 2 AssHt I,J (M), gradeM p = t.
