On the Graded Injective Hull of R/𝔭 in Prime Characteristic

Let [Formula: see text] be a polynomial ring in [Formula: see text] indeterminates with coefficients in a field [Formula: see text] of characteristic [Formula: see text] and [Formula: see text] be the ring of differential operators over [Formula: see text] We consider [Formula: see text] as a nonmaximal homogeneous prime ideal, and [Formula: see text] as a graded injective hull of [Formula: see text] In this paper, we prove the generalized Eulerian property for the graded injective hull [Formula: see text] as a [Formula: see text]-module in positive characteristic. Furthermore, we show that the minimal injective resolution of [Formula: see text]-finite module ends in an object which is the direct sum of indecomposable in graded case using some finiteness properties in prime characteristic.