Let M be a graded leftA-module and M∗ the associate complex of M. Then : If is noetherian (resp. artinian) then strongly hopfian cohopfian); cohopfian), leftA-module, M, N submodule N∗ fully invariant subcomplex M∗. M∗/N∗ hopfian, hopfian. if all cohopfian, cohopfian.

The concept of a 2-Absorbing submodule is considered as an essential feature in the field module theory and has many generalizations. This articale discusses Extend Nearly Pseudo Quasi-2-Absorbing submodules their relationship to submodule, Nearly-2-Absorbing Pseudo-2-Absorbing rest other concepts previously studied. between them been studied, explaining that opposite not true under certain con...

We consider the Arveson–Douglas conjecture on the essential normality of homogeneous submodules corresponding to algebraic subvarieties of the unit ball. We prove that the property of essential normality is preserved by isomorphisms between varieties, and we establish a similar result for maps between varieties that are not necessarily invertible. We also relate the decomposability of an algebr...

We will determine the structure of the modular standard modules of association schemes of class two. In the process, we will give the theoretical interpretation for the p-rank theory for strongly regular graphs, and understand the p-rank as the dimension of a submodule of the modular standard module. Considering the modular standard module, we can obtain the detailed classification more than th...

The modular multilevel converter (MMC) is receiving extensive research interests in high/medium voltage applications due to its modularity, scalability, reliability, high voltage capability and excellent harmonic performance. The submodule capacitors are usually quite bulky since they have to withstand fundamental frequency voltage fluctuations. To reduce the capacitance of these capacitors, th...

let $r$ be an arbitrary ring and $t$ be a submodule of an $r$-module $m$. a submodule $n$ of $m$ is called $t$-small in $m$ provided for each submodule $x$ of $m$, $tsubseteq x+n$ implies that $tsubseteq x$. we study this mentioned notion which is a generalization of the small submodules and we obtain some related results.

let $m_r$ be a module with $s=end(m_r)$. we call a submodule $k$ of $m_r$ annihilator-small if $k+t=m$, $t$ a submodule of $m_r$, implies that $ell_s(t)=0$, where $ell_s$ indicates the left annihilator of $t$ over $s$. the sum $a_r(m)$ of all such submodules of $m_r$ contains the jacobson radical $rad(m)$ and the left singular submodule $z_s(m)$. if $m_r$ is cyclic, then $a_r(m)$ is the unique ...

The purpose of this paper is to investigate pure submodules of multiplication modules. We introduce the concept of idempotent submodule generalizing idempotent ideal. We show that a submodule of a multiplication module with pure annihilator is pure if and only if it is multiplication and idempotent. Various properties and characterizations of pure submodules of multiplication modules are consid...

The complement operation on modules has not been argued so far as we know. We observe that it plays an important role in software maintenance, as well as join and meet operations. A de nition of a module and operations on them is given in category theoretic terms; we adopt well-known characterization of complement as a left adjoint to join. The de nition is rst given locally and then extended t...

‎A module $M$ is called FI-extending if every fully invariant submodule of $M$ is essential in a direct summand of $M$‎. ‎It is not known whether a direct summand of an FI-extending module is also FI-extending‎. ‎In this study‎, ‎it is given some answers to the question that under what conditions a direct summand of an FI-extending module is an FI-extending module?