Generalizations of Nakayama ring. III
[For part II see the review above.] Using the same definitions and notations as in the above paper, the author continues his study of right Artinian rings having (*,n) or (**,n). In the first section a right Artinian ring with (*,1) for any hollow module is discussed. It is observed that if \(J^ 2=0\), then R satisfies (*,2) and hence (*,1). If R is an algebra over an algebraically closed field \(K_ 0\), then R satisfies II'': \(eRe/eJe=\bar eK'\) for each primitive idempotent e, K' a field in the center of R. It is shown that for an algebra R over a field K with condition II'', the following are equivalent: 1) R satisfies (*,2) and \(eJ=A_ 1\oplus B_ 1\) such that \(A_ 1/J(A_ 1)\not\approx B_ 1/J(B_ 1)\), \(A_ 1\), \(B_ 1\) hollow. 2) R satisfies (*,n) for all n. In the above paper US-3 rings with (*,1) or (*,2) were studied. Now US-4 algebras over an algebraically closed field are observed. Provided that \(J^ 3=0\) \((J^ 2\neq 0)\) all lattices of submodules in eR are given for each primitive idempotent e. In the final section R is an algebra over a field with condition II''. Necessary and sufficient conditions are given in order that R be a US-4 algebra with (*,1) in terms of the structure of eR (e a primitive idempotent). With a slight modification a similar structure theorem is obtained for (*,2).
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