On Notherian Dimension of Artinian modules with uniserial dimension

In this article we present a direct proof of the fact show that every Artinian uniserial (i.e., its submodules are linearly ordered by induction) module over commutative or right Noetherian rings are 1-atomic. Consequently, we show that if R is a commutative (or, a right Noetherian) ring, then the Noetherian dimension of an Artinian module with the uniserial dimension is less than or equal to 1