Krull dimension in serial rings (Q581629)

Let R be a right and left serial ring with right Krull dimension \(\alpha\). The author shows that R has left Krull dimension \(\alpha\). This is done by showing that \(J(\alpha)^ k=0\) for some k. Here, \(J(\alpha)\) is the ideal of R defined by \[ J(\alpha)= \begin{cases} \text{Jacobson radical of R,} &\text{ if \(\alpha =0\)} \\ \cap^{\infty}_{n-1}J(\alpha -1)^ n, &\text{ if \(\alpha\) is a nonlimit ordinal} \\ \cap_{\beta <\alpha}J(\beta), &\text{ if \(\alpha\) is a limit ordinal.}\end{cases} \] The author defines and makes extensive use of what she terms \(\alpha\)- cliques of local projectives. These \(\alpha\)-cliques are based on \textit{R. B. Warfield}'s notion [J. Algebra 37, 187-222 (1975; Zbl 0319.16025)] of successor and predecessor, and are avowed to play a significant role in future work on the structure of serial rings with Krull dimension.