Geodesic
Bi-artinian noetherian rings
Glasgow mathematical journal, Tome 43 (2001) no. 1, pp. 9-21

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A noetherian ring R satisfies the descending chain condition on two-sided ideals (“is bi-artinian”) if and only if, for each prime P ∈ spec(R), R/P has a unique minimal ideal (necessarily idempotent and left-right essential in R/P). The analogous statement for merely right noetherian rings is false, although our proof does not use the full noetherian condition on both sides, requiring only that two-sided ideals be finitely generated on both sides and that R/Q be right Goldie for each Q ∈ spec(R). Examples exist, for each n∈N and in all characteristics, of bi-artinian noetherian domains Dn with composition series of length 2n and with a unique maximal ideal of height n. Noetherian rings which satisfy the related E-restricted bi-d.c.c. do not, in general, satisfy the second layer condition (on either side), but do satisfy the Jacobson conjecture. 
Whelan, E. A. Bi-artinian noetherian rings. Glasgow mathematical journal, Tome 43 (2001) no. 1, pp. 9-21. doi: 10.1017/S0017089501010023
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