A Note on Pure Submodule Relative to Submodule

In this paper, the primary objective is to obtain decomposition theorems for graded modules over the polynomial ring $k[x]$, where $k$ denotes a field. There is some overlap with recent work of Höppner and Lenzing. The results obtained include identification of the free, projective, and injective modules. It is proved that a module that is either reduced and locally finite or bounded below is a direct sum of cyclic submodules. Pure submodules are direct summands if they are bounded below. In such case, the pure submodule is itself a direct sum of cyclic submodules. It is also noted that Cohen and Gluck’s Stacked Bases Theorem remains true if the modules are graded.

Let R be a commutative ring with identity 1 and M be a unitary left R-module. A submodule N of an R-module M is said to be approximately pure submodule of an R-module, if for each ideal I of R. The main purpose of this paper is to study the properties of the following concepts: approximately pure essentialsubmodules, approximately pure closedsubmodules and relative approximately pure complement submodules. We prove that: when an R-module M is an approximately purely extending modules and N be Ap-puresubmodulein M, if M has the Ap-pure intersection property then N is Ap purely extending.

An R-module M is called strongly FP-injective if for any finitely presented R-module P and any i > 0. Denoted by the class of all strongly FP-injective R-modules and by the left orthogonal class with respect to Comparing with some classical results of Noetherian rings, we show that a ring is coherent if and only if is closed under pure submodules; if and only if any finitely generated module in is finitely presented; if and only if R is -coherent and is closed under direct sums; if and only if for any nonnegative integer n, any finitely presented left R-module X, any R-T-bimodule M and any injective right T-module E. In addition, we show that a module is injective if and only if it is a pure -periodic module, where denotes the class of all injective modules.

In this paper, we introduce and study the pure-direct-projective modules, that is the modules [Formula: see text] every pure submodule [Formula: see text] of which with [Formula: see text] isomorphic to a direct summand of [Formula: see text] is a direct summand of [Formula: see text]. We characterize rings over which every right [Formula: see text]-module is pure-direct-projective. We examine for which rings or under what conditions pure-direct-projective right [Formula: see text]-modules are direct-projective, projective, quasi-projective, pure-projective, flat or injective. We prove that over a Noetherian ring every injective module is pure-direct-projective and a right hereditary ring [Formula: see text] is right Noetherian if and only if every injective right [Formula: see text]-module is pure-direct-projective. We obtain some properties of pure-direct-projective right [Formula: see text]-modules which have [Formula: see text] and [Formula: see text].