Let 1$"> X = ( ∑ n = 1 ∞ ℓ 1 n ) p , p > 1 . In this paper, we investigate M -ideals which are also ideals in L ( X ) , the algebra of all bounded linear operators on X . We show that K ( X ) , the ideal of compact operators on X is the only proper closed ideal in L ( X ) which is both an ideal and an M -ideal in L ( X ) .