%0 Journal Article %T On the Noetherian dimension of Artinian modules with homogeneous uniserial dimension %J Bulletin of the Iranian Mathematical Society %I Iranian Mathematical Society (IMS) %Z 1017-060X %A Alehafttan, A.R. %A Shirali, N. %D 2017 %\ 12/30/2017 %V 43 %N 7 %P 2457-2470 %! On the Noetherian dimension of Artinian modules with homogeneous uniserial dimension %K Noetherian dimension %K homogeneous uniserial dimension %K atomic modules %R %X  ‎In this article‎, ‎we first‎ ‎show that non-Noetherian Artinian uniserial modules over‎ ‎commutative rings‎, ‎duo rings‎, ‎finite $R$-algebras and right‎ ‎Noetherian rings are $1$-atomic exactly like $\Bbb Z_{p^{\infty}}$‎. ‎Consequently‎, ‎we show that if $R$ is a right duo (or‎, ‎a right‎ ‎Noetherian) ring‎, ‎then the Noetherian dimension of an Artinian‎ ‎module with homogeneous uniserial dimension is less than or equal‎ ‎to $1$‎. ‎In particular‎, ‎if $A$ is a quotient finite dimensional‎ ‎$R$-module with homogeneous uniserial dimension‎, ‎where $R$ is a‎ ‎locally Noetherian (or‎, ‎a Noetherian duo) ring‎, ‎then $n$-dim ‎$A\leq‎ ‎1$‎. ‎We also show that the Krull dimension of Noetherian modules is‎ ‎bounded by the uniserial dimension of these modules‎. ‎Moreover‎, ‎we introduce the concept of qu-uniserial modules and by using this‎ ‎concept‎, ‎we observe that if $A$ is an Artinian $R$-module‎, ‎such that‎ ‎any of its submodules is qu-uniserial‎, ‎where $R$ is a right duo (or‎, ‎a right Noetherian) ring‎, ‎then $n$-dim $‎A\leq 1$. %U http://bims.iranjournals.ir/article_1146_2706ad754c4976e7ef148b92f0a82975.pdf