Artinian ring
The ring "R" can be considered as a right module, where the action is the natural one given by the ring multiplication on the right. "R" is called right Artinian when this right module "R" is an Artinian module. The definition of "left Artinian ring" is done analogously. For noncommutative rings this distinction is necessary, because it is possible for a ring to be Artinian on one side only.