Generalizations of Nakayama ring. II (Q1087973)

In part I [ibid. 23, 181-200 (1986; Zbl 0588.16018)] the author has introduced conditions (*,n) and (**,n) as generalizations of a serial ring. In this paper the condition: (*,3): every (non-zero) maximal submodule of a direct sum D(3) of 3 nonzero hollow modules is also a direct sum of hollow modules is studied. Maximal submodules of a direct sum of hollow modules were studied by the author in a number of papers. It turned out that another condition was needed to get some structure theorems, i.e. \(| eJ/eJ^ 2| \leq 2\) for each primitive idempotent e, where J is the Jacobson radical of R. If a right Artinian ring R has (*,3) and the above condition, an earlier result can be used to obtain the same structure theorem. In this paper it is shown that for Artinian rings R with \(| eJ/eJ^ 2| \leq 2\), each e, the following are equivalent: 1) every submodule of any D(3) is a direct sum of hollow modules, 2) (*,3) holds for any D(3), 3) eR has the structure above for each primitive idempotent e. Right US-3 rings were defined as rings satisfying (**,3). In this paper the structure of right US-3 rings is investigated assuming (*,1) or (*,2). [For part III see the following review.]