arXiv subjects

Nasrin Shirali

Publications and source records attributed to Nasrin Shirali.

4 recordsLinked to original sources

arXiv2026

Weakly uniserial dimension of modules

Recall that a module is called weakly uniserial if its submodules are comparable regarding embedding. Weakly uniserial modules are a nontrivial generalization of uniserial modules. In this paper we define and study a new dimension, which measure how far a module deviates from being weakly uniserial. We call this dimension, weakly uniserial dimension. Also, we define and study monoartinian (mononoetherian) modules. We say that an $R$-module $M$ is monoartinian (mononoetherian) if in every descending (ascending) chain of submodules of $M$, except probably a finite number, each module in chain embedded in the next (previous) one. We show that a module has weakly uniserial dimension if and only if it is monoartonian.

math.RA↗

arXiv2023

A topological expression for dual-classical Krull dimension of rings

Let R be a ring and X = SH(R)-{0} be the set of all non-zero strongly hollow ideals (briefly, sh-ideals) of R. We first study the concept SH-topology and investigate some of the basic properties of a topological space with this topology. It is shown that if X is with SH-topology, then X is Noetherian if and only if every subset of X is quasi-compact if and only if R has dcc on semi-sh-ideals. Finally, the relation between the dual-classical Krull dimension of R and the derived dimension of X with a certain topology has been studied. It is proved that, if X has derived dimension, then R has the dual-classical krull dimension and in case R is a D-ring (i.e., the lattice of ideals of R is distributive), then the converse is true. Moreover these two dimension differ by at most 1.

math.GN↗

arXiv2023

On AB5* modules with Noetherian dimension

In this paper, we study the Noetherian dimension of sum of certain modules. It is proved that for any module M which is an irredundant sum of submodules, each of which has Noetherian dimension less than alpha, if M has finite spanning dimension (fsd-module, for short) or it is a weakly atomic module, then Noetherian dimension M less than alpha. Here, by a weakly atomic module we mean a module M for which every proper non-small submodule N, has Noetherian dimension strictly less than that of M. Also, it is proved that if M is an AB5* module with Noetherian dimension and N_i is a family of submodules of M such that Noetherian dimension M over N_i, less than alpha, for each i, then Noetherian dimension M over intersection of N_is less than alpha. Using this, we give a structure theorem for alpha-short modules in the category of AB5* and finally, we classify alpha-short modules in this category.

math.RA↗

arXiv2023

On multiplication fs-modules and dimension symmetry

In this paper, we first study $fs$-modules, i.e., modules with finitely many small submodules. We show that every $fs$-module with finite hollow dimension is Noetherian. Also, we prove that an $R$-module $M$ with finite Goldie dimension, is an $fs$-module if and only if $M = M_1 \oplus M_2$, where $M_1$ is semisimple and $M_2$ is an $fs$-module with $Soc(M_2) \ll M$. Then, we investigate multiplication $fs$-modules over commutative rings and show that $R$ is an $fs$-ring if and only if every multiplication $R$-module is an $fs$-module. In particular, we prove that the lattices of $R$-submodules of $M$ and $S$-submodules of $M$ are coincide, where $S=End_R(M)$. Consequently, $M_R$ and $_SM$ have the same dimension of Krull (Noetherian, Goldie and hollow). Further, we prove that for any self-generator multiplication module $M$, to be an $fs$-module as a right $R$-module and as a left $S$-module are equivalent.

math.RA↗