In this paper we will study artinian quotients A = R/I of the polynomial ring R = k[x1, ..., xr], where k is a field of characteristic zero, the xi’s all have degree 1 and I is a homogeneous ideal of R. These rings are often referred to as standard graded artinian algebras. Before explaining the main results of this work, we establish some of the notation we will use: the h-vector of A is h(A) ...

We show  that a projective maximal submodule of afinitely generated, regular, extending module is a directsummand. Hence, every finitely generated, regular, extendingmodule with projective maximal submodules is semisimple. As aconsequence, we observe that every regular, hereditary, extendingmodule is semisimple. This generalizes and simplifies a result of  Dung and   Smith. As another consequen...