RINGS WITH INVOLUTION AND CHAIN-CONDITIONS

The structure of involution rings with d.c.c. and a.c.c. on *-biideals is investigated. If an involution ring A has d.c.c. on *-biideals, then its Jacobson radical is nilpotent, and A is an artinian ring with artinian radical. If an involution ring has a.c.c. on *-biideals, then its Baer radical is finitely generated as an abelian group. For a polynomial ring A[x] over a nonassociative involution ring A a criterion is given to satisfy a.c.c. on *-biideals. In particular, a polynomial ring A[x] over an associative involution ring A has a.c.c. on *-biideals if and only if A is finite and semiprime; this characterization can be considered as an involutive counterpart of the Hilbert Basis Theorem. These results are valid also for rings without involution, and in this way (commutative) rings with a.c.c. on biideals are characterized. Also examples are given for disproving some expectations.