On the Noetherian dimension of Artinian modules with homogeneous uniserial dimension

‎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 Zp∞‎. ‎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≤‎‎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≤1.