When is there a nontrivial extension-closed subcategory?

Let R be a commutative Noetherian local ring, and denote by mod R the category of finitely generated R-modules. In this paper, we consider when mod R has a nontrivial extension-closed subcategory. We prove that this is the case if there are part of a minimal system of generators x,y of the maximal ideal with xy=0, and that it holds if R is a stretched Artinian Gorenstein local ring which is not a hypersurface.