LET C BE A SEMIDUALIZING R-MODULE. WE PRESENT SOME NECESSARY AND SUFFICIENT CONDITIONS FOR C TO BE DUALIZING IN SEVERAL DIRECTIONS. FIRST, WE SHOW THAT VANISHING OF APPROPRIATE BASS NUMBERS (RESP. DUAL BASS NUMBERS) ON THE CLASS FC (RESP. IC) IS EQUIVALENT WITH THE DUALIZING PROPERTY OF C. NEXT, WE SHOW THAT C IS DUALIZING IF AND ONLY IF THE CLASS IC IS CLOSED WITH RESPECT TO TORSION PRODUCT. FINALLY, WE DEFINE TWO NEW NOTIONS, NAMELY N-C-PERFECT AND N-C-COPERFECT MODULES, AND GIVE A CHARACTERIZATION FOR DUALIZING MODULES IN THE CASE WHERE (R, M) IS A DDIMENSIONAL LOCAL RING, AS FOLLOWS: C IS DUALIZING IF AND ONLY IF E (R/M) (RESP. R B) IS D-C-PERFECT (RESP. D-C-COPERFECT).

LET I BE AN IDEAL OF A COMMUTATIVE NOETHERIAN RING RAND M BE AN R -MODULE. FOR A ZD-MODULE M WE PROVE THAT IF HI I(M) IS A ZD-MODULE FOR ALL 0 £ I<T, THEN HOMR (R=I; HT I(M)) IS A ZD-MODULE.

LET R BE A RIGHT ARTINIAN RING OR A PERFECT COMMUTATIVE RING. LET M BE A NONCOSINGULAR LIFTING MODULE THAT DOES NOT HAVE RELATIVELY PROJECTION COMPONENT. THEN (FORMULA) HAS THE EXCHANGE PROPERTY AND THE DECOMPOSITION COMPLEMENTS DIRECT SUMMANDS, WHERE EACH ENDOMORPHISM RING END (MI) IS LOCAL

LET (R, M) BE A NOETHERIAN LOCAL RING WHICH IS A QUOTIENT OF A LOCAL GORENSTEIN RING. LET M BE AN R-MODULE WITH A FINITELY GENERATED SUBMODULE L SUCH THAT M/L IS ARTINIAN. IT IS SHOWN THAT THE DEFICIENCY MODULES OF MATLIS DUAL OF M IS ISOMORPHIC TO THE DEFICIENCY MODULES OF MATLIS DUAL OF M/L.

LET R BE A RING WITH UNITY AND M BE A UNITARY LEFT R-MODULE. THE INTERSECTION GRAPH OF AN R-MODULEM, DENOTED BY G(M), IS DEFINED TO BE A GRAPH WHOSE VERTICES ARE IN ONE TO ONE CORRESPONDENCE WITH ALL NON-TRIVIAL SUB MODULES OF M AND TWO DISTINCT VERTICES ARE ADJACENT IF AND ONLY IF THE CORRESPONDING SUB MODULES OFM HAVE NON-ZERO INTERSECTION. IN THIS TALK, WE STUDY ARTINAN MODULES, NOETHERIAN MODULES AND INJECTIVE MODULES, WHOSE INTERSECTION GRAPHS ARE COMPLETE. IN ADDITION, FOR A NOETHERIAN R-MODULE M, WITH COMPLETE INTERSECTION GRAPH, WE GIVE A CONDITION UNDER WHICH M IS ARTINIAN.

FOR A FINITELY GENERATED MODULE M OVER A LOCAL RING (A,M), LET RB(M):=ťN=0 BNM (RESP. GB(M):= ťN=0M/BN+1M) DENOTE THE REES MODULE OF M ASSOCIATED TO THE IDEAL B OF A (RESP. THE ASSOCIATED GRADED MODULE OF M WITH RESPECT TO B). IN THIS TALK WE STUDY SOME PROPERTIES OF SUCH MODULES. SOME INDEPENDENCE RESULTS ABOUT THE REDUCTION NUMBER OF B RELATIVE TO M WILL BE INVESTIGATED. WE ALSO PROVE THAT FOR G=GRADE(G(B)+,GB(M)) < MIN{GRADE(B,M), L(B,M)} THE INEQUALITY AG(GB(M))<AG+1(GB(M)) HOLDS, WHERE G(B)+ DENOTES THE IRRELEVANT IDEAL OF ASSOCIATED GRADED RING G(B), _(B,M) IS THE ANALYTIC SPREAD OF B RELATIVE TO M AND AI(GB(M)) IS THE END OF GRADED MODULE HIG(B)+ (GB(M)) FOR I=G, G+1.