Note on left serial algebras (Q1090400)

Let R be a left and right artinian ring with identity. Let (*,n) be the condition: every maximal submodule of the direct sum of n arbitrary R- hollow modules is also a direct sum of hollow modules. The author proves the following Theorem: Let R be a left serial ring. Then R satisfies (*,1) if eJ is a direct sum of uniserial modules for each primitive idempotent e (J the Jacobson radical). Let R be an algebra over a field K with the following condition: (A) \(eRe/eJe=eK+eJe\) for each primitive idempotent e. Theorem. Let R be a left serial algebra with (A) and put \(J(eR)=\sum^{n(e)}_{i=1}\oplus A_ i\), \(J(A_ i)=\sum^{n_ i}_{j=1}\oplus B_{ij}\), where the \(A_ i\) and \(B_{ij}\) are hollow. Assume that \(J^ 4=0\). The following are equivalent: 1) R satisfies (*,1). 2) eR has the following structure: If \(\bar B{}_{ij}\approx C_{i'j'}\), then \(B_{ij}\) is uniserial, where \(\bar B{}_{ij}=B_{ij}/B_{ij}J\) and \(C_{i'j'}\) is a simple submodule in \(J(B_{i'j'})\), (i\(\neq i')\).