On the fissility of semiprimary rings (Q793823)

Let A be an associative ring (not necessarily with identity). The author calls A fissile if the maximal torsion ideal of A is a ring-direct summand of A. If A is semiprimary, the author calls an idempotent e of A principal if \(e+J\) is the identity of A/J, where J denotes the Jacobson radical of A. Suppose that A is a semiprimary ring and that e is a principal idempotent of A such that \(ann e=0.\) The author shows that A is fissile and, if A/J is torsionfree, he shows that A is divisible. The main result of the paper follows from this; namely, if e is a principal idempotent of a semiprimary ring A, then A contains a fissile subring B such that \(A=B\oplus ann e\) and \(ann e\subseteq J.\)