HUCKABA AND KELLER SAY A COMMUTATIVE RING R HAS PROPERTY (A) IF EVERY FINITELY GENERATED IDEAL OF R CONSISTING ENTIRELY OF ZERO DIVISORS HAS NONZERO ANNIHILATOR. HONG ET AL. EXTEND PROPERTY (A) TO NONCOMMUTATIVE RINGS AND SHOW THAT NEITHER RIGHT SELF-INJECTIVE RINGS NOR VON NEUMANN REGULAR RINGS HAVE PROPERTY (A), IN GENERAL. WE CONTINUE IN THIS PAPER THE STUDY OF RINGS WITH PROPERTY (A), AND CONSIDER CONDITIONS UNDER WHICH RIGHT FP-INJECTIVE RINGS, MININJECTIVE RINGS, STRONGLY RIGHT AB RINGS AND RINGS WITH QUASI-IFP HAVE PROPERTY (A).

LET R BE A COMMUTATIVE RING AND LET M BE AN R-MODULE. DENOTE BY ZR(M) THE SET OF ALL ZERO-DIVISORS OF R ON M. M IS CALLED STRONGLY PRIMAL (RESP. SUPER PRIMAL) IF FOR ARBITRARY A, B Î ZR(M) (RESP. EVERY FINITE SUBSET F OF ZR(M)) THE ANNIHILATOR OF {A, B} (RESP. F) IN M IS NON-ZERO. IN THIS PAPER WE GIVE SOME RESULTS ON THESE CLASSES OF MODULES. ALSO WE PROVIDE A RELATIONSHIP AMONG THE FAMILIES OF PRIMAL, STRONGLY PRIMAL AND SUPER PRIMAL MODULES.

LET R BE A RING, M A MONOID AND S: M®END (R) A MONOID HOMOMORPHISM. A FUNDAMENTAL CONSTRUCTION IN NON-COMMUTATIVE RING THEORY IS THE FORMATION OF THE SKEW MONOID RING R*M. THE UTILITY OF THIS CONSTRUCTION LIES IN THE FACT THAT THIS RINGS IS IMPORTANT IN CONSTRUCTING INTERESTING MATHEMATICAL OBJECTS AND THEY HAVE PROVIDED MANY IMPORTANT (COUNTER) EXAMPLES IN NON-COMMUTATIVE ALGEBRA. THIS PAPER CONSIDERS THE PROBLEM OF DETERMINING WHEN A SKEW MONOID R*M OVER R IS EITHER NILPOTENT OR A ZERO-DIVISOR.

LET R BE A RING, S A STRICTLY ORDERED MONOID AND W: S® END (R) A MONOID HOMOMORPHISM. IN [4], MARKS, MAZUREK AND ZIEMBOWSKI STUDY THE CLASS OF (S,W) -ARMENDARIZ RINGS, AS A GENERALIZATION OF THE STANDARD ARMENDARIZ CONDITION FROM ORDINARY POLYNOMIAL RING TO SKEW GENERALIZED POWER SERIES RING. WE OBSERVE FROM RESULTS IN [4], THAT THE UPPER NILRADICAL COINCIDES WITH THE PRIME RADICAL IN (S, W) -ARMENDARIZ RINGS AND ALSO EVERY ONE-SIDED NIL IDEAL OF THESE RINGS IS CONTAINED IN A TWO-SIDED NIL IDEAL OF THE RING, NAMELY SATISFIES IN THE KOTHE’S CONJECTURE. ALSO IT CAN BE SHOWN THAT THE FACTOR RINGS OF AN (S, W) -ARMENDARIZ RINGS OVER ITS PRIME RADICAL IS ALSO (S, W) -ARMENDARIZ. WE CONTINUE IN THIS PAPER THE STUDY OF RINGS WITH SUCH PROPERTY IN SKEW GENERALIZED POWER SERIES RINGS AND BRING SOME PROPERTIES OF THESE RINGS.

LET R BE A COMMUTATIVE RING AND M BE AN R-MODULE, IF (FORMULA) AND (FORMULA). THE ANNIHILATOR GRAPH OF MODULE M OVER RING R IS THE GRAPH AG(RM) WITH VERTICES (FORMULA) AND TWO VERTICES X AND Y ARE ADJACENT IF AND ONLY IF ANN (IXIYM) ¹(FORMULA). IT FOLLOWS THAT IF M BE A FAITFUL R-MODULE, THEN EACH EDGE OF G (RM) IS AN EDGE OF AG (RM). WE SHOW THAT IF M BE A MULTIPLICATION MODULE THEN AG (RM) IS CONNECTED WITH DIAMETER AT MOST THREE AND WITH GIRTH AT MOST FOUR PROVIDED THAT AG (RM) HAS A CYCLE.