An Artinian ring is defined to be a ring that is both left Artinian and right Artinian. So satisfies Descending chain condition (DCC) on left and right ideals. This implies it also satisfies DCC for the set of 2-sided ideals. But the converse isn’t true, that satisfying the DCC condition on 2-sided ideals means it satisfies it for left and right ideals. So is there a notion of “2-sided Artinian” with the DCC condition only on 2sided ideals?
I can’t find any reference to it or what it would be used for.