If N is a submodule of the R-module M , and a ∈ R, let λa : M/N → M/N be multiplication by a. We say that N is a primary submodule of M if N is proper and for every a, λa is either injective or nilpotent. Injectivity means that for all x ∈ M , we have ax ∈ N ⇒ x ∈ N . Nilpotence means that for some positive integer n, aM ⊆ N , that is, a belongs to the annihilator of M/N , denoted by ann(M/N). ...

We introduce a notion of pattern occurrence that generalizes both classical permutation patterns as well as poset containment. Many questions about pattern statistics and avoidance generalize naturally to this setting, and we focus on functional complexity problems – particularly those that arise by constraining the order dimensions of the pattern and text posets. We show that counting the numb...

Let R be a ring with identity and C(R) denote the category of complexes of R-modules. In this paper we study the homotopy categories arising from projective (resp. injective) complexes as well as Gorenstein projective (resp. Gorenstein injective) modules. We show that the homotopy category of projective complexes over R, denoted K(Prj C(R)), is always well generated and is compactly generated p...

Proposition 1. M is injective if and only if its singular value decomposition M = UDV H has a V that is square and invertible. In this case, MM is invertible and M = (MHM)−1MH . Proof. Let M be an r × c matrix. Suppose that M is injective, so that rank(M) = c because the kernal is zero. Then D is a c × c matrix and so V H is also c× c. V H must already be injective (lest M not be injective), an...

α1 α2 α3 α4 inModR. If the rows are exact, then the following statements hold. (1.3.a) If γ2 and γ4 are injective and γ1 is surjective, then γ3 is injective. (1.3.b) If γ2 and γ4 are surjective and γ5 is injective, then γ3 is surjective. Proof. First we prove (1.3.a). So suppose that γ2 and γ4 are injective and γ1 is surjective. Start with m3 ∈ M3 with the property that γ3(m3) = 0. The goal is ...