Rings With Comparability

The class of rings studied in this paper properly contains the class of right distributive rings which have at least one completely prime ideal in the Jacobson radical. Amongst other results we study prime and semiprime ideals, right noetherian rings with comparability and prove a structure theorem for rings with comparability. Several examples are also given.

Serial Right Noetherian Rings

A module M is called a serial module if the family of its submodules is linearly ordered under inclusion. A ring R is said to be serial if RR as well as RR are finite direct sums of serial modules. Nakayama [8] started the study of artinian serial rings, and he called them generalized uniserial rings. Murase [5, 6, 7] proved a number of structure theorems on generalized uniserial rings, and he described most of them in terms of quasi-matrix rings over division rings. Warfield [12] studied serial both sided noetherian rings, and showed that any such indecomposable ring is either artinian or prime. He further showed that a both sided noetherian prime serial ring is an (R:J)-block upper triangular matrix ring, where R is a discrete valuation ring with Jacobson radical J. In this paper we determine the structure of serial right noetherian rings (Theorem 2.11).

A note on Jacobson's conjecture for right Noetherian rings

In 1956, Jacobson asked whether the intersection of the powers of the Jacobson radical, J(R), of a right Noetherian ring R, must always be zero [4, p. 200]. His question was answered in the negative by I. N. Herstein [3], who noted that , where Z(2) denotes the ring of rational numbers with denominator prime to 2, affords a counterexample. In contrast, the ring , though similar in appearance to R1, satisfies . (Here, k denotes a field.)