Quasi-ideals and bi-ideals in radical theory

*—Bi—ideals in Radical Theory of Involution Rings

“…Now, let W be *-bistrong and B4bA with B E W. Hence B@BoP<A@AoP where * is the exchange involution. By Proposition 1, BOP E W and since B @ B O P is an extension of B by BOP, we have B @ B O P E W. Consequently It has been proved in Proposition 8 of[5] that for any radical class W the conditions "left and right stable", "bistable" and "bistrong" are equivalent. In view of this result and Propositions 2 and 3, we deduce the following theorem.…”

“…The relationship between quasi-ideals (bi-ideals) and radical classes of associative rings was introduced and studied by Stewart and Wiegandt [5]. Throughout this note, we extend this study to the radical classes of involution rings.…”

“…If B a: A, B E W then B C W(A) for every ring A with involution.In [4], Sands has proved that a left hereditary and left strong radical class is also right hereditary and right strong. Proposition 8 in[5] implies that such a class is also bihereditary. However, this class need not be bistrong, consider for example the Jacobson radical.…”