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, th en its Jacobson radical is nilpotent, and A is an artinian ring with a rtinian radical. If an involution ring has a.c.c. on -biideals, then its Baer radical is finitely generated as an abelian group. For a poly nomial ring A[x] over a nonassociative involution ring A a criterion i s given to satisfy a.c.c. on -biideals. In particular, a polynomial r ing A[x] over an associative involution ring A has a.c.c. on -biideal s if and only if A is finite and semiprime; this characterization can be considered as an involutive counterpart of the Hilbert Basis Theore m. These results are valid also for rings without involution, and in t his way (commutative) rings with a.c.c. on biideals are characterized. Also examples are given for disproving some expectations.