Primary-like and weakly primary-like submodules are two new generalizations of primary ideals from rings to modules. In fact, the class of primary-like submodules of a module lie between primary submodules and weakly primary-like submodules properly.  In this note, we show that these three classes coincide when their elements are submodules of a multiplication module and satisfy the primeful pr...

Let $N$ be a submodule of a module $M$ and a minimal primary decomposition of $N$ is known‎. ‎A formula to compute Baer's lower nilradical of $N$ is given‎. ‎The relations between classical prime submodules and their nilradicals are investigated‎. ‎Some situations in which semiprime submodules can be written as finite intersection of classical prime submodule are stated‎.

In this article, we develop the tool of saturation in the context of primary-like submodules of modules. We are particularly interested in relationships among the saturation of a primary-like submodule satisfying the primeful property and its radical. Furthermore, we provide sufficient conditions involving saturation and torsion arguments under which the radical of such a submodule is prime.

a module $m$ is called $emph{h}$-cofinitely supplemented if for every cofinite submodule $e$ (i.e. $m/e$ is finitely generated) of $m$ there exists a direct summand $d$ of $m$ such that $m = e + x$ holds if and only if $m = d + x$, for every submodule $x$ of $m$. in this paper we study factors, direct summands and direct sums of $emph{h}$-cofinitely supplemented modules. let $m$ be an $emph{h}$...

A module $M$ is called $emph{H}$-cofinitely supplemented if for every cofinite submodule $E$ (i.e. $M/E$ is finitely generated) of $M$ there exists a direct summand $D$ of $M$ such that $M = E + X$ holds if and only if $M = D + X$, for every submodule $X$ of $M$. In this paper we study factors, direct summands and direct sums of $emph{H}$-cofinitely supplemented modules. Let $M$ be an $emph{H}...

Let C be an algebraically closed field of positive characteristic p and complete with respect to a non-archimedean absolute value | . | and Λ ⊂ C a discrete Fp-submodule. Suppose there exists an Fp-basis {λ0, λ1, . . .} of Λ such that 0 < |λ0| < |λ1| < · · · and |λi| −→ ∞. For k ∈ N define the meromorphic function Ck,Λ(z) = ∑ λ∈Λ 1 (z − λ)k on C. We show that all the zeroes x of Ck,Λ satisfy (∗...

The main goal of this article is to explore the concepts graded ϕ-2-absorbing and primary submodules as a new generalization 2-absorbing submodules. Let ϕ:GS(M)→GS(M)⋃{∅} be function, where GS(M) denotes collection R-submodules M. A proper K∈GS(M) said R-submodule M if whenever x,y are homogeneous elements R s element with xys∈K−ϕ(K), then xs∈K or ys∈K xy∈(K:RM), we call K xs ys in radical xy∈(...

Let R be ring and M a right R-module. This article introduces the concept of τ −⊕-supplemented modules as follows: Given a hereditary torsion theory in Mod-R with associated torsion functor τ we say that a module M is τ −⊕-supplemented when for every submodule N of M there exists a direct summand K of M such that M = N +K and N ∩K is τ−torsion, and M is called completely τ −⊕-supplemented if ev...

Let $p$ be a prime, let $K$ discretely valued extension of $\mathbb{Q}_p$, and $A_{K}$ an abelian $K$-variety with semistable reduction. Extending work by Kim Marshall from the case where $p>2$ $K/\mathbb{Q}_p$ is unramified, we prove $l=p$ complement Galois cohomological formula Grothendieck for $l$-primary part N\'eron component group $A_{K}$. Our proof involves constructing, each $m\in \math...