The Matrix Ring under Kronecker Product: Balancing Noetherian and Artinian Properties in Algebraic Structures

الملخص

 In the matrix ring under the Kronecker product, Noetherian properties are typically observed due to the strong algebraic structure. Ideals are finitely generated, and ascending chains stabilize after a finite number of steps. However, its classification as Artinian is often hindered by the complexities in ideal representation and the stability of descending chains. These challenges may lead to instability, making it less likely for the structure to exhibit Artinian properties. While the matrix ring under the Kronecker product reflects Noetherian characteristics due to its structural and operational framework, it remains less representative of Artinian properties due to its added complexities.