Notion of 2-sided artinian ring?

Question

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.

Edit: But for the converse, having DCC on 2-sided ideals give DCC on left (or right) ideals, seems nontrivial to me. Since there can be many more left ideals than 2-sided 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.

Answer 1

So is there a notion of "2-sided Artinian" with the DCC condition only on 2sided ideals?

I think the typical thing to do is to say "DCC on two-sided ideals," without using the "Artinian" adjective. I see lots of hits for this sort of thing when I search.

But for the converse, having DCC on 2-sided ideals give DCC on left (or right) ideals, seems nontrivial to me. Since there can be many more left ideals than 2-sided ideals..

Of course, your intuition is right it is not true that DCC on two-sided ideals would imply DCC on left ideals. For example, you can take any non-Artinian (or non-Noetherian) simple ring. It would necessarily have only two ideals, but infinite descending (resp. ascending and descending!) chains of left ideals.

Answer 2

I believe my confusion was in the terminology of the definition: where it says "descending chain condition on left and right ideals" it means DCC on 2-sided ideals, rather than DCC on all left ideals and DCC on all right ideals...?