PRIME IDEALS IN RINGS WITH INVOLUTION

摘要

LetRbe a ring with involution *. We show that ifRis a *-prime ring which is not a prime ring, thenRis #x201C;essentially#x201D; a direct product of two prime rings. Moreover, ifPis a *-prime *-ideal ofR, which is not a prime ideal ofR, andXis minimal among prime ideals ofRcontainingP, thenPis a prime ideal ofX,P=X#x2229;X* and either: (1)Pis essential inXandXis essential inR;or (2) for any relative complementCofPinX, thenC* is a relative complement ofXinR. Further characterizations of *-primeness are provided.