Consider the space R∆ of rational functions of r variables with poles on an arrangement of hyperplanes ∆. It is important to study the decomposition of the space R∆ under the action of the ring of differential operators with constant coefficients. In the one variable case, a rational function of z with poles at most on z = 0 is written uniquely as φ(z) = Princ(φ)(z)+ψ(z) where Princ(φ)(z) = ∑ n...

Let R be a commutative ring with identity. Let φ : I(R) → I(R) ∪ {∅} be a function where I(R) denotes the set of all ideals of R. A proper ideal Q of R is called φ-primary if whenever a, b ∈ R, ab ∈ Q−φ(Q) implies that either a ∈ Q or b ∈ √ Q. So if we take φ∅(Q) = ∅ (resp., φ0(Q) = 0), a φ-primary ideal is primary (resp., weakly primary). In this paper we study the properties of several genera...

A finite module M over a noetherian local ring R is said to be Gorenstein if Ext(k, M) = 0 for all i 6= dimR. A endomorphism φ : R → R of rings is called contracting if φ(m) ⊆ m for some i ≥ 1. Letting R denote the R-module R with action induced by φ, we prove: A finite R-module M is Gorenstein if and only if HomR( R,M) ∼= M and ExtiR( R,M) = 0 for 1 ≤ i ≤ depthR.

The proof of Theorem 2 in the main body of the paper applies to the special case in which the covariance matrix of random vector (θ; θM ;u) is full rank. In this Online Appendix, we prove Theorem 2 for the general case, imposing only Assumptions 1 and 2L: Cov(v, θ|θM ) 6= 0 and V ar(u|θ, θM ) > 0. Proof of Theorem 2 (General Case) Before proceeding to the proof, we first prove the following lem...

Let $(R,underline{m})$ be a commutative Noetherian local ring and $M$ be a non-zero finitely generated $R$-module. We show that if $R$ is almost Cohen-Macaulay and $M$ is perfect with finite projective dimension, then $M$ is an almost Cohen-Macaulay module. Also, we give some necessary and sufficient condition on $M$ to be an almost Cohen-Macaulay module, by using $Ext$ functors.