On order rings of semi-primary rings (Q1175584)

Let \(R\) be an order in a semi-primary ring \(Q\). If \(R\) satisfies the maximal condition on nil one sided ideals of \(R\) then the author shows that an ideal \(I\) has finite length as a left module if and only if it has finite length as a right module. As a consequence, the artinian radical, \(A(R)\), is well-behaved. In the case, when one can arbitrarily re-order the prime ideals in any product \(P_ 1\dots P_ n=0\), it follows that \(A(R/N)=A(R)+N/N\), where \(N\) is the nil radical of \(R\).