IRANIAN ALGEBRA SEMINAR | Year:2009 | دوره:20 |Papers Count:73

LET G BE A 3-GROUP OF MAXIMAL CLASS OF ORDER 3N. IN THIS TALK WE GIVE A STRUCTURE THEOREM FOR THE FULL AUTOMORPHISM GROUP OF G.

IN THIS NOTE, THE NOTIONS OF DUAL RIGHT AND DUAL LEFT STABILIZERS OF A SET IN BOUNDED BCK-ALGEBRAS ARE DEFINED AND THE RELATIONSHIP BETWEEN OF THEM ARE INVESTIGATED. THEN WE DEFINE A CLASS OF SPECIAL BOUNDED BCK-ALGEBRAS CALLED DUAL NORMAL BCK-ALGEBRAS. FINALLY WE PROVE THAT THE DUAL SEMISIMPLE BOUNDED BCK-ALGEBRAS AND DUAL J-SEMISIMPLE BOUNDED BCK-ALGEBRAS ARE ALL DUAL NORMAL’S.

WE INVESTIGATE THE CLASS OF PIECEWISE PRIME, PWP, A RING WHICH PROPERLY INCLUDES ALL PIECEWISE DOMAIN. FOR A PWP RING WE DETERMINE A LARGE CLASS OF RING EXTENSIONS WHICH HAVE A GENERALIZED TRIANGULAR MATRIX REPRESENTATION FOR WHICH THE DIAGONAL RINGS ARE PRIME. WE INVESTIGATE THE QUASI-BAER AND RELATED CONDITIONS ON MONOID RINGS R[G]. IN PARTICULAR WE CONSIDER THE TRANSFER OF THE QUASI-BAER AND RELATED CONDITIONS BETWEEN R AND A U.P.-MONOID RING R[G] OF A U.P.-MONOID G OVER R. WE CHARACTERIZE THE SEMI-CENTRAL IDEMPOTENTS OF VARIOUS RING EXTENSIONS OF R IN TERMS OF THE SEMI-CENTRAL IDEMPOTENTS OF R.

IN ANY FINITE GROUP G, THE COMMUTATIVITY DEGREE OF G (DENOTED BY D(G)) IS THE PROBABILITY THAT TWO RANDOMLY CHOSEN ELEMENTS OF G COMMUTE. THE AIM OF THIS PAPER IS TO GENERALIZE DEFINITION OF D(G) FOR EVERY COMPACT GROUP G (INFINITE AND EVEN UNCOUNTABLE). WE SHALL STATE SOME RESULTS CONCERNING THE COMPACT GROUP WHICH ARE MOSTLY NEW OR IMPROVEMENTS OF KNOWN RESULTS GIVEN IN [4].

WE INTRODUCE TWO EQUIVALENCE RELATIONS ON SEMIHYPERGROUPS. ALSO WE DISCUSS ON SOME PROPERTIES OF THESE TWO RELATIONS AND INVESTIGATE ON QUOTIENT SEMIHYPERGROUPS VIA THESE RELATIONS.

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.

CASTELNUOVO-MUMFORD REGULARITY IS A KIND OF UNIVERSAL BOUND FOR IMPORTANT INVARIANTS OF GRADED ALGEBRAS, SUCH AS THE MAXIMUM DEGREE OF THE SYZYGIES AND THE MAXIMUM NON-VANISHING DEGREE OF THE LOCAL COHOMOLOGY MODULES. IN THIS PAPER WE GIVE A SIMPLE CRITERION IN TERMS OF REES ALGEBRA OF A GIVEN IDEAL TO SHOW THAT HIGH ENOUGH POWERS OF THIS IDEAL HAVE LINEAR RESOLUTION.IN THIS TALK WE DISCUSS ABOUT UPPER BOUND FOR REGULARITY OF POWERS OF AN IDEAL AND SOME IMPLEMENTATIONS IN COCOA ARE ALSO INVOLVED.

A SIMPLE UNDIRECTED GRAPH IS SAID TO BE SEMISYMMETRIC IF IT IS REGULAR AND EDGE-TRANSITIVE BUT NOT VERTEX-TRANSITIVE. LET P ³ 11 BE A PRIME. IN THIS PAPER, IT IS PROVED THAT, EVERY CUBIC EDGE-TRANSITIVE ELEMENTARY ABELIAN REGULAR COVER OF Q3 IS VERTEX-TRANSITIVE.

IN THIS PAPER THE CONCEPTS OF A FUZZY PRIMARY SUBACT IS GIVEN AND SOME FUNDAMENTAL RESULTS ARE PROVED. ALSO A CHARACTERIZATION OF A FUZZY P-PRIMARY SUBACT IS GIVEN.

LET T BE A COMPACTLY GENERATED TRIANGULATED CATEGORY WHICH IS CLOSED UNDER ARBITRARY SMALL COPRODUCTS. BASED ON A CONSTRUCTION OF LOCAL COHOMOLOGY FUNCTORS ON TRIANGULATED CATEGORIES, WHICH IS INTRODUCED RECENTLY BY BENSON, IYENGAR AND KRAUSE, A NOTION OF SUPPORT IS ASSIGNED TO ANY OBJECT X OF T. USING THIS NEW NOTION OF SUPPORT, WE WILL STUDY LOCAL COHOMOLOGY FUNCTORS AND SHOW THAT IN SPECIAL CASES, OUR RESULTS RECOVER AND GENERALIZE THE KNOWN RESULTS ON THE USUAL LOCAL COHOMOLOGY FUNCTORS.

LET R BE A NOETHERIAN RING, AN IDEAL OF R SUCH THAT DIM R/A=1 AND M A FINITE R–MODULE. WE WILL STUDY COFINITENESS AND SOME OTHER PROPERTIES OF THE LOCAL COHOMOLOGY MODULES HIA(M).

AN UNDIRECTED GRAPH WITHOUT ISOLATED VERTICES SAID TO BE SEMISYMMETRIC IF ITS FULL AUTOMORPHISM GROUP ACTS TRANSITIVELY ON ITS EDGE SET BUT NOT ON ITS VERTEX SET. IN THIS PAPER WE PROVE THAT FOR EVERY PRIME P, THERE IS NO SEMISYMMETRIC CUBIC GRAPH OF ORDER 4P3.

AN R-MODULE M IS CALLED SEMIPRIME IF IT IS COGENERATED BY EACH OF ITS ESSENTIAL SUBMODULES. WE INVESTIGATE WHEN A SEMIPRIME MODULE M HAS THE PROPERTY THAT FOR EVERY NON-ZERO SUBMODULE N£M, THERE EXISTS AN R-HOMOMORPHISM Q:M®N SUCH THAT Q(N)¹0. IT IS PARTIALLY SOLVED AN OPEN PROBLEM POSED IN [2].

GENERALIZED RELATIVE INJECTIVITY WAS RENAMED AS OJECTIVE BY MOHAMED AND MULLER [4]. THEY ALSO STUDIED THE CONCEPT OF OJECTIVE MODULES IN TWO WAYS, NAMELY TO COJECTIVE AND *COJECTIVE MODULES [5]. HERE WE INTRODUCE THE CONCEPT OF M-PJECTIVITY, WHICH IS A GENERALIZATION OF M−*COJECTIVITY, AND GIVE SOME RELATIONS BETWEEN PJECTIVE AND LIFTING MODULES. WE ALSO STUDY THE CONCEPT OF GENERALIZED LIFTING MODULES, AND GIVE SOME PROPERTIES OF SUCH MODULES.

LET R =ÄN³0 RN BE A STANDARD GRADED RING, A0 AN IDEAL OF THE BASE RING R0 AND LET M BE A NON-ZERO FINITELY GENERATED GRADED RMODULE. IN THIS TALK, WE STUDY THE ASYMPTOTIC BEHAVIOUR OF THE SEQUENCE (GRADE (A0, HIR+ (M)N))NÎZ, OF GRADES OF COMPONENTS OF THE I-TH GRADED LOCAL COHOMOLOGY MODULE, WHEN N®˫¥, IN THE FOLLOWING CASES:(I) I = FR+(M);(II) I = GR+(M);(III) DIM(R0)£2.TO THIS END WE STUDY ARTINIAN AND TAMENESS PROPERTIES OF CERTAIN GRADED LOCAL COHOMOLOGY MODULES.

LET G BE A P-GROUP OF MAXIMAL CLASS OF ORDER PN. IT IS SHOWN THAT THE ORDER OF THE GROUP OF ALL AUTOMORPHISMS OF G CENTRALIZING THE FRATTINI QUOTIENT TAKES THE MAXIMUM VALUE P2N-4 IF AND ONLY IF G IS METABELIAN.

WE GIVE SEVERAL NECESSARY AND SUFFICIENT CONDITIONS FOR THE EXISTENCE OF THE PRESENTATION BY CONJUGATION FOR A REDUCED EXTENDED AFFINE WEYL GROUP. WE INVENT A COMPUTATIONAL TOOL BY WHICH ONE CAN DETERMINE SIMPLY IF A GIVEN EXTENDED AFFINE WEYL GROUP HAS SUCH A PRESENTATION. AS AN APPLICATION, WE DETERMINE THE EXISTENCE OF THE PRESENTATION BY CONJUGATION FOR A LARGE CLASS OF EXTENDED AFFINE WEYL GROUPS.

BY A QUASI-PERMUTATION MATRIX WE MEAN A SQUARE MATRIX OVER THE COMPLEX FIELD C WITH NON-NEGATIVE INTEGRAL TRACE. THUS EVERY PERMUTATION MATRIX OVER C IS A QUASI-PERMUTATION MATRIX. FOR A GIVEN FINITE GROUP G, THE MINIMAL DEGREE OF A FAITHFUL REPRESENTATION OF G BY QUASI-PERMUTATION MATRICES OVER THE RATIONAL AND THE COMPLEX NUMBERS ARE DENOTED BY Q (G) AND C (G) RESPECTIVELY. FINALLY R (G) DENOTES THE MINIMAL DEGREE OF A FAITHFUL RATIONAL VALUED COMPLEX CHARACTER OF G. THE PURPOSE OF THIS PAPER IS TO CALCULATE C (G) AND R (G) FOR THE MAXIMAL PARABOLIC SUBGROUPS OF CHEVALLEY GROUPS.

LET D BE A DIVISION RING. AN EASY CONSEQUENCE OF A FAMOUS RESULT OF HERESTEIN ASSERTS THAT IF D* (THE MULTIPLICATIVE GROUP OF D) IS AN FC GROUP (GROUP WITH FINITE CONJUGACY CLASSES), THEN IT IS ABELIAN. A SIMILAR RESULT IS ALSO TRUE FOR ANY FC SUBNORMAL SUBGROUP N OF D* (SEE LEMMA 1 OF BELOW). NOW, LET M BE A MAXIMAL SUBGROUP OF N. IN THIS TALK, I PROVE THAT IF M IS AN FC GROUP, THEN IT IS ABELIAN.

THROUGHOUT ALL RINGS ARE COMMUTATIVE WITH IDENTITY AND ALL MODULES ARE UNITARY. DEPENDING ON AN R-MODULE M, RAD(M) IS AN R-LATTICE OF RADICAL SUBMODULES OF M AND Z (M) IS A ZARISKI SPACE OVER THE SEMIRING Z (R). THERE ARE TWO ISOMORPHIC FULL SUBCATEGORIES RAD AND ZAR, WHOSE OBJECTS ARE THE SEMIMODULES RAD(M) AND Z (M) OVER THE SIMIRING ID (R) OF THE IDEALS OF R. THESE PROVIDE TWO NATURALLY ISOMORPHIC FUNCTORS R AND Z FROM MOD TO SEMOD RESPECTIVELY. BESIDE THE INVESTIGATION PRESERVING AND REFLECTING EXACT SEQUENCES BY R AND Z, IT IS PROVED THAT R AND Z PRESERVE FINITE FREE REPRESENTATIONS OF MODULES OVER A DOMAIN R.

IN THIS TALK, WE STUDY BEHAVIORS OF BAER INVARIANTS AND VARIETAL COVERING GROUPS WITH RESPECT TO DIRECT LIMITS. THE MAIN RESULT HAS A USEFUL APPLICATION IN ORDER TO EXTEND SOME KNOWN STRUCTURES OF VARIETAL COVERING GROUPS FOR SEVERAL FAMOUS PRODUCTS OF FINITELY MANY TO AN ARBITRARY FAMILY OF GROUPS.

WE SHALL CALL A MONOID S PRINCIPALLY WEAKLY (WEAKLY) LEFT COHERENT IF DIRECT PRODUCTS OF NON-EMPTY FAMILIES OF PRINCIPALLY WEAKLY (WEAKLY) FLAT RIGHT S-ACTS ARE PRINCIPALLY WEAKLY (WEAKLY) FLAT. SUCH MONOIDS HAVE NOT BEEN STUDIED IN GENERAL. HOWEVER, BULMAN FLEMING AND MCDOWELL PROVED THAT A COMMUTATIVE MONOID S IS (WEAKLY) COHERENT IF AND ONLY IF THE ACT SI IS WEAKLY FLAT FOR EACH NON-EMPTY SET I. IN THIS PAPER WE INTRODUCE THE NOTION OF FINITE (PRINCIPAL) WEAK FLATNESS FOR CHARACTERIZING (PRINCIPALLY) WEAKLY LEFT COHERENT MONOIDS. ALSO WE INVESTIGATE MONOIDS OVER WHICH DIRECT PRODUCTS OF ACTS TRANSFER AN ARBITRARY FLATNESS PROPERTY TO THEIR COMPONENTS.

IN [2] IT WAS NOTED THAT IN COMMUTATIVE RINGS WITH ZERO DIVISORS THE EQUIVALENT DEFINITIONS OF ASSOCIATES IN INTEGRAL DOMAINS ARE NOT EQUIVALENT ANYMORE AND THUS LEAD TO DIFFERENT TYPES OF IRREDUCIBLE ELEMENTS. THERE IT WAS PROVED THAT IN SUCH RINGS THESE DIFFERENT TYPES OF ASSOCIATES ARE EQUIVALENT IF AND ONLY IF THE RING, R, IS PRESIMPLIFIABLE, THAT IS FOR EVERY R, AÎR, RA=A IMPLIES A = 0 OR RÎU(R). IN [3] THEY EXTENDED THE BASIC DEFINITIONS OF THE THEORY OF FACTORIZATION TO MODULES; INCLUDING PRESIMPLIFIABLITY. HERE WE DEFINE THE WEAKLY PRESIMPLIFIABLE CONDITION FOR MODULES. THIS NOTION ENABLES US TO REDUCE CHECKING PRESIMPLIFIABLITY AND SEVERAL OTHER FACTORIZATION PROPERTIES IN GENERAL MODULES TO CHECKING THEM JUST IN FAITHFUL MODULES. ALSO WE STUDY HOW THESE PROPERTIES BEHAVE UNDER SEVERAL MODULE CONSTRUCTIONS.

IN THIS PAPER, WE DEFINE COVER OF AN S-POSET A, COESSENTIAL EPIMORPHISM AND PROJECTIVE COVER IN POS-S. ALSO WE SHOW THAT PROJECTIVE COVER IN POS-S IS DETERMINED UNIQUELY UP TO ISOMORPHISM AND THAT PROJECTIVE COVERS OF S-POSETS DO NOT ALWAYS EXIST.

THIS PAPER IS DEVOTED TO STUDY THE STRUCTURE OF THE NONABELIAN TENSOR PRODUCT OF LIE ALGEBRAS AND THE CONNECTION OF NILPOTENCY AND SOLVABILITY BETWEEN THE LIE ALGEBRAS AND THEIR TENSOR PRODUCTS. IN ADDITION, SOME CERTAIN BOUNDS FOR THE ORDER AND THE TENSOR PRODUCTS AND THE SCHUR MULTIPLIER OF NILPOTENT LIE ALGEBRAS WILL BE PRESENTED.

LET D BE AN ARBITRARY DIVISION RING. WE DENOTE BY MN(D) AND PN(D) THE SET OF ALL N × N MATRICES AND THE SET OF ALL N × N IDEMPOTENT MATRICES OVER D RESPECTIVELY. IN THIS TALK WE DISCUSS ABOUT CHARACTERIZATIONS OF COMMUTATIVITY PRESERVING LINEAR MAPS ON MN(D) AND COMMUTATIVITY PRESERVING (NOT NECESSARILY LINEAR) MAPS ON PN(D).

DISTINCT NONLINEAR COMPLEX IRREDUCIBLE CHARACTERS OF A GROUP NEED NOT TO HAVE DISTINCT KERNELS. IN THIS PAPER, WE CONSIDER FINITE P-GROUPS WITH THE PROPERTY THAT NONLINEAR COMPLEX IRREDUCIBLE CHARACTERS HAVE DISTINCT KERNELS. THE AIM OF THIS NOTE IS TO CHARACTERIZE P-GROUPS WITH THIS PROPERTY CONTAINING AT MOST THREE NONLINEAR IRREDUCIBLE COMPLEX CHARACTERS.

LET G BE A NON-CYCLIC P-GROUP OF ORDER PN, N ³3. IT HAS BEEN CONJECTURED THAT |G| £ |AUT(G)P|, WHERE AUT (G)P DENOTES A SYLOW P-SUBGROUP OF AUT(G). IN THIS TALK WE CLASSIFY THE FINITE NON-ABELIAN METACYCLIC P-GROUPS G, P > 2, FOR WHICH |G| = |AUT(G)P|. IN PARTICULAR WE GIVE AN ANSWER TO AN EXTENDED VERSION OF A PROBLEM POSED BY Y. BERKOVICH.

THE COMMUTATIVITY DEGREE OF GROUPS AND RINGS HAS BEEN STUDIED BY CERTAIN AUTHORS SINCE 1973, AND THE MAIN RESULT OBTAINED IS PR(A)£ 5/8 , WHERE PR(A) IS THE COMMUTATIVITY DEGREE OF A NON-ABELIAN GROUP (OR RING) A. VERIFYING THIS INEQUALITY FOR AN ARBITRARY SEMIGROUP A IS A NATURAL QUESTION AND IN THIS PAPER BY PRESENTING AN INFINITE CLASS OF FINITE NONCOMMUTATIVE SEMIGROUPS WE PROVE THAT THE COMMUTATIVITY DEGREE MAY BE ARBITRARILY CLOSE TO 1. WE NAME THIS CLASS OF SEMIGROUPS THE ALMOST COMMUTATIVE OR APPROXIMATELY ABELIAN SEMIGROUPS.

LET G BE A GRAPH OF ORDER N AND LET Μ BE AN EIGENVALUE OF MULTIPLIC- ITY M. A STAR COMPLEMENT FOR Μ IN G IS AN INDUCED SUBGRAPH OF G OF ORDER N-M WITH NO EIGENVALUE Μ. HERE WE FIRST IDENTIFY AMONG CACTUS GRAPHS, COMPLETE THREE PARTITE GRAPHS KN,N,N, AND BICYCLIC GRAPHS WHICH CAN BE STAR COMPLEMENT FOR 1 AS THE SECOND LARGEST EIGENVALUE. USING THE GRAPHS OBTAINED, WE NEXT SEARCH FOR THEIR MAXIMAL EXTENSIONS, EITHER BY THEORETICAL MEANS, OR BY COMPUTER AIDED SEARCH.

ACCORDING TO Z. LIU AND R. ZHAO [5] A RING R IS CALLED WEAK ARMENDARIZ IF WHENEVER POLYNOMIALS F(X) = A0 + A1X + · · · + AMXM, G(X) = B0 + B1X + · · · + BNXN Î R[X] SATISFY F(X) G(X) = 0, THEN AIBJ 2 NIL(R) FOR ALL I, J. THEY HAVE SHOWN THAT, IF R IS A SEMICOMMUTATIVE RING, THEN THE RING R[X] AND THE RING R[X]/(XN) , WHERE (XN) IS THE IDEAL GENERATED BY XN, AND N IS A POSITIVE INTEGER, ARE WEAK ARMENDARIZ. IN THIS NOTE WE INTRODUCE WEAK ARMENDARIZ IDEALS WHICH ARE A GENERALIZATION OF IDEALS HAVE THE WEAKLY INSERTION OF FACTORS PROPERTY (OR SIMPLY WEAKLY IFP) AND INVESTIGATE THEIR PROPERTIES.

IT IS KNOWN THAT ALTHOUGH THE BAER CRITERION FOR INJECTIVITY HOLDS FOR MODULES OVER RINGS WITH UNIT, IT IS NOT TRUE FOR ACTS OVER AN ARBITRARY MONOID. RECENTLY, THE PRESENT AUTHOR, TOGETHER WITH M. EBRAHIMI AND M. MAHMOUDI, PUBLISHED A PAPER (COMM. ALGEBRA 35 (2007), 3912- 3918) GIVING FOR THE ACTS OVER SOME CLASSES OF SEMIGROUPS, THE BAER CRITERION IS TRUE. IN THIS PAPER WE FIND ANOTHER CLASSE OF MONOIDS SUCH THAT FOR ACTS OVER THEM THE BAER CRITERION HOLDS.

LET G BE A GROUP AND LET S BE A SUBSET OF G SUCH THAT 1GÏS AND S=S−1. LET G=CAY(G, S) BE A CAYLEY GRAPH ON G RELATIVE TO S. THEN G IS SAID TO BE NORMAL EDGE-TRANSITIVE, IF NAUT(G)(G) ACTS TRANSITIVELY ON EDGES. IN THIS PAPER WE DETERMINE ALL NORMAL EDGE-TRANSITIVE CAYLEY GRAPHS ON A QUATERNION GROUP Q2N OF VALENCY 4. IN ADDITION WE GIVE NORMAL EDGE-TRANSITIVE CAYLEY GRAPHS ON A QUATERNION GROUP Q2N OF SOME OTHER VALENCIES.

IN THIS TALK, WE INVESTIGATE THE M × N FUZZY LINEAR SYSTEM WHOSE COEFFICIENTS MATRIX IS CRISP AND RIGHT-HAND SIDE COLUMN IS A FUZZY NUMBER VECTOR. WE FIRST REPLACE THE ORIGINAL M × N FUZZY LINEAR SYSTEM BY A (2M) × (2N) CRISP FUNCTION LINEAR SYSTEM. AND THEN THE FUZZY LEAST SQUARES SOLUTIONS TO THE SYSTEM ARE DISCUSSED BY USING THE HUANG’S METHOD. THE METHOD IS DISCUSSED IN DETAIL AND ILLUSTRATED BY SOLVING NUMERICAL EXAMPLE.

IN THIS PAPER WE GIVE A GENERALIZATION OF A RESULT OF J.A. FRASER AND W.K. NICHOLSON [2]. LET R BE A RING AND R BE A WEAKLY RIGID AUTOMORPHISM OF R. IF R IS SKEW POWER SERIES ARMENDARIZ, THEN R[[X;A]] IS RIGHT PRINCIPALLY PROJECTIVE IF AND ONLY IF R IS RIGHT PRINCIPALLY PROJECTIVE AND ANY COUNTABLE FAMILY OF IDEMPOTENTS IN R HAS A GENERALIZED JOIN WHEN ALL LEFT SEMICENTRAL IDEMPOTENTS ARE CENTRAL.

THROUGHOUT THIS TALK, ALL RINGS CONSIDERED WILL BE COMMUTATIVE NOETHERIAN AND WILL HAVE NON-ZERO IDENTITY ELEMENTS. SUCH A RING WILL BE DENOTED BY R AND A TYPICAL IDEAL OF R WILL BE DENOTED BY I. LET K BE A NON-ZERO FINITELY GENERATED MODULE OVER R. THE CONCEPT OF WEAKLY GK-PERFECT IDEAL WAS INTRODUCED BY GOLOD IN [5]. HE SHOWED THAT, THIS NEW IDEAL HAS SOME NICE PROPERTIES. FOR INSTANCE, HE PROVED THAT FOR A WEAKLY GK-PERFECT IDEAL I OF R, K IS A CANONICAL MODULE FOR R IF AND ONLY IF EXTSR (R/I,K) IS A CANONICAL MODULE FOR R/I, WHERE S=GRADEKI.

LET M BE A MODULE. THEN M IS CALLED A D11 MODULE IF ANY SUBMODULE OF M HAS A SUPPLEMENT WHICH IS A DIRECT SUMMAND OF M. ALSO M IS CALLED D+11 IF EVERY DIRECT SUMMAND OF M IS D11. IN THIS PAPER WE INVESTIGATE GENERALIZATIONS OF D11 AND D+11 MODULES, NAMELY D-D11 AND D-D+11 MODULES. WE WILL PROVE THAT ANY D-D+11 MODULE M HAS A DECOMPOSITION M=M1ÅM2 WITH D<(M1) <<D M1 AND D(M2)=M2.

LET G BE A SIMPLE UNDIRECTED GRAPH AND LET DG BE A SIMPLICIAL COMPLEX WHOSE FACES CORRESPEND TO THE INDEPENDENT SETS OF G. WE SHOW THAT COMPLETE T-PARTITE GRAPH G IS SEQUENTIALLY COHEN-MACAULAY IF AND ONLY IF G IS SHELLABLE. ALSO WE SHOW THAT COMPLETE T-PARTITE GRAPH G IS VERTEX DECOMPOSABLE IF AND ONLY IF A1 BE ARBITRARY AND AI = 1 FOR ALL 2 £ I £ T.

THE NOTION OF PRIME FUZZY SUBNEXUS OF A NEXUS IS DEFINED, AND SOME RELATED RESULTS ARE OBTAINED. IN PARTICULAR BY CONSIDERING THE CONCEPT OF HOMOMORPHISM SOME THEOREMS ABOUT THE COIMAGE AND PREIMAGE ARE PROVED.

IN THIS TALK I SHALL REPORT ON OUR JOINT ONGOING WORK WITH GIANNI LANDI AND WALTER VAN SUIJLEKOM. WE DEFINE A NOTION OF HOLOMORPHIC STRUCTURE IN TERMS OF A BIGRADING OF A SUITABLE DIFFERENTIAL CALCULUS OVER THE QUANTUM SPHERE. REALIZING THE QUANTUM SPHERE AS A PRINCIPAL HOMOGENEOUS SPACE OF THE QUANTUM GROUP SUQ(2) PLAYS AN IMPORTANT ROLE IN OUR APPROACH. WE DEFINE A NOTION OF HOLOMORPHIC VECTOR BUNDLE AND ENDOW THE CANONICAL LINE BUNDLES OVER CP1Q WITH A HOLOMORPHIC STRUCTURE. WE ALSO DEFINE THE QUANTUM HOMOGENEOUS COORDINATE RING OF CP1Q AND IDENTIFY IT WITH THE COORDINATE RING OF THE QUANTUM PLANE. FINALLY I SHALL FORMULATE AN ANALOGUE OF CONNES’ THEOREM, CHARACTERIZING HOLOMORPHIC STRUCTURES ON COMPACT ORIENTED SURFACES IN TERMS OF POSITIVE CURRENTS, IN OUR NONCOMMUTATIVE CONTEXT. THE NOTION OF TWISTED POSITIVE HOCHSCHILD COCYCLE PLAYS AN IMPORTANT ROLE HERE.

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.

THIS NOTE, THE NOTIONS OF AN EXPANSION OF FILTERS AND (T,S)-PRIMARY FILTERS ARE INTRODUCED, AND RELATED PROPERTIES ARE INVESTIGATED.

IN THIS PAPER WE FIND THE ORDER AND STRUCTURE OF AUTC(G) FOR G = K H BY IMPOSING SOME CONDITIONS ON H AND K. AS A RESULT, WE SHOW THAT IF M IS A NON-NORMAL MAXIMAL SUBGROUP OF A SOLVABLE GROUP G, SUCH THAT ALL CENTRAL AUTOMORPHISMS OF G FIX COREG(M) POINTWISE, THEN AUTC(G), IS META-ABELIAN.

GRAPH IS HALF-TRANSITIVE IF ITS AUTOMORPHISM GROUP ACTS TRANSITIVELY ON ITS VERTEX SET AND EDGE SET, BUT NOT ON ITS ARC SET. LET P BE A PRIME. CHAO [ON THE CLASSIFICATION OF SYMMETRIC GRAPHS WITH A PRIME NUMBER OF VERTICES, TRANS. AMER. MATH. SOC. 158 (1971) 247-256] PROVED THAT THERE ARE NO HALF-TRANSITIVE GRAPHS ON P VERTICES. BY CHENG AND OXLEY [ON WEAKLY SYMMETRIC GRAPHS OF ORDER TWICE A PRIME, J. COMBIN. THEORY B 42 (1987) 196-211], ALSO THERE ARE NO HALF-TRANSITIVE GRAPHS OF ORDER 2P. IN THIS PAPER AN EXTENSION OF THE ABOVE RESULTS IN THE CASE OF TETRAVALENT GRAPHS IS GIVEN. IT IS PROVED THAT THERE ARE NO TETRAVALENT HALF-TRANSITIVE GRAPHS OF ORDER 2P2.

THIS TALK, WE SHOW THAT IF G= ZPA1 N * ZPA2 N * … N * ZPA2T IS &NBSP;THE NTH NILPOTENT PRODUCT OF SOME CYCLIC P-GROUPS, WHERE &NBSP;C1 ³ N,A1 ³&NBSP; A2 ³ … ³ AT AND (Q, P)=1 FOR ALL PRIME Q LESS THAN OR EQUAL TO N, THEN |NC1, C2, ..., CS M(G)|= PDM IF AND ONLY IF G=ZPN* ZPN* … N*ZP, WHERE M =STI=1AI AND DM=XCS+1 … (XC2+1 (SNJ XC1+J (M) …).

LET G BE A GROUP. A SUBSET N OF G IS SAID TO BE NON-COMMUTING IF XY¹YX FOR ANY TWO DISTINCT ELEMENTS X AND Y IN N. IF ÇNdz ÇMÇ FOR ANY OTHER NON-COMMUTING SUBSET M IN G, THEN N IS SAID TO BE A MAXIMAL NON-COMMUTING SUBSET. IN THIS PAPER WE OBTAIN LOWER BOUNDS FOR THE CARDINALITY OF A MAXIMAL NON-COMMUTING SUBSET, BY N-SINGER GENERATOR, IN A THREE-DIMENSIONAL GENERAL LINEAR GROUP. MOREOVER WE OBTAIN STRUCTURAL INFORMATION ABOUT EVERY MAXIMAL NON-COMMUTING SUBSET CONTAINING NO PROPER POWERS.

FOR A GIVEN FINITE GROUP G WE CONSTRUCT A TRANSITIVE PERMUTATION GROUP GG WHOSE RELATED ASSOCIATION SCHEME INV (GG) IS COMMUTATIVE. WE INVESTIGATE SOME RELATIONSHIP BETWEEN THE GROUP THEORETICAL PROPERTIES OF G AND THE COMBINATORIAL THEORETICAL PROPERTIES OF INV (GG). FURTHERMORE, THE CHARACTER TABLE OF THE ASSOCIATION SCHEME INV (GG) IS COMPUTED.

IN THIS PAPER, WE INTRODUCE THE NOTION OF PSEUDO VALUATIONS ON A BCK-AL-GEBRA AND STUDY ITS PROPERTIES. WE INVESTIGATE THE RELATION BETWEEN PSEUDO VALUATIONS AND IDEALS. WE USE PSEUDO VALUATION TO DEFINE A PSEUDO METRIC ON A BCK-ALGEBRA AND PROVE THAT THIS PSEUDO METRIC INDUCES A CONGRUENCE RELATION ON A BCK-ALGEBRA. WE DEFINE THE QUOTIENT ALGEBRA INDUCED BY THIS RELATION AND PROVE THAT IT IS ALSO A BCK-ALGEBRA.

A GROUP G IS CALLED AN E-GROUP IF THE NEAR-RING GENERATED BY THE ENDOMORPHISMS OF G IN THE NEAR-RING OF MAPS ON G IS A RING. IT IS WELL KNOWN (SEE, E.G., MALONE, 1995) THAT A GROUP G IS AN E-GROUP IF AND ONLY IF EACH ELEMENT COMMUTES WITH ITS ENDOMORPHIC IMAGES. FOR ANY PRIME NUMBER P, WE CALL AN E-GROUP WHICH IS ALSO A P-GROUP, A PE-GROUP. IN THIS PAPER AT FIRST WE EXPLAIN GENERAL PROPERTIES OF E-GROUPS. ALSO WE PROVE THAT AN INFINITE FINITELY GENERATED E-GROUP IS THE DIRECT PRODUCT OF A CENTRAL TORSION-FREE SUBGROUP AND A FINITE SUBGROUP. NEXT, WE PROVE THAT THERE IS NO 3E-GROUP OF NILPOTENCY CLASS 3 OF ORDER AT MOST 310. ALSO WE CONSTRUCT A GROUP OF CLASS 3 WHICH IS “VERY CLOSE” TO BE AN E-GROUP. THE FOLLOWING QUESTIONS ARE CENTRAL ONES IN THIS PAPER: (1) WHAT IS THE LEAST NUMBER OF GENERATORS OF A FINITELY GENERATED NONABELIAN E-GROUP?(2) WHAT IS THE MINIMUM ORDER OF A FINITE NON-ABELIAN PE-GROUP?WE PROVE THAT THE MINIMAL NUMBER OF GENERATORS OF A FINITELY GENERATED NON-ABELIAN E-GROUP IS 4.IN RESPONSE TO THE QUESTION (2), WE PROVE THAT THE MINIMUM ORDER OF A FINITE NON-ABELIAN PE-GROUP IS P8, FOR ANY ODD PRIME NUMBER P AND THIS ORDER IS 27 FOR P=2.ALSO WE OBTAIN A NEW CLASS OF E-GROUPS.AS WE HAVE FOUND THAT SOME OF OUR RESULTS ARE VALID FOR A VERY LARGER CLASS OF FINITE P-GROUPS THAN PE-GROUPS, WE STUDY A CLASS OF P-GROUPS FOR EVERY PRIME NUMBER P AND WE DENOTE THIS CLASS OF P-GROUPS BY PE. (A FINITE P-GROUP G IS CALLED A PE-GROUP IF G IS A 2-ENGEL GROUP AND ALL ELEMENTS OF ORDER AT MOST PR LIE IN THE CENTER OF G, WHERE PR IS EXPONENT G/G1).WE CLASSIFY ALL 3-GENERATOR PE-GROUPS AND PE-GROUPS WITH CYCLIC DERIVED SUBGROUP AND DETERMINE ENDOMORPHISMS OF 3-GENERATOR PE-GROUPS AND PE-GROUPS. FINALLY WE CLASSIFY ALL PE-GROUPS AND PE-GROUPS OF ORDER AT MOST P7 FOR ANY PRIME NUMBER P.

IN THIS MANUSCRIPT, WE SHOW THAT IF H H= AÄB, THEN H/A IS ISOMORPHIC TO B, WHERE H IS A HYPER K-ALGEBRA, A AND B ARE CLOSED SETS.

THE B-PARTS OF REAL NUMBERS WERE INTRODUCED AND STUDIED IN [4], AND B-ADDITION OF REAL NUMBERS IN [3]. THE B-PARTS HAVE MANY INTERESTING NUMBER THEORETIC EXPLANATIONS, ANALYTIC AND ALGEBRAIC PROPERTIES AND THE B-ADDITION IS B-DECIMAL PART OF ORDINARY ADDITION OF TWO REAL NUMBERS AND IS DENOTED +B. IT WAS SHOWN THAT (R,+B) IS A SEMIGROUP (EQUIVALENTLY THE B-DECIMAL PART FUNCTION ( )B SATISFIES THE ASSOCIATIVE FUNCTIONAL EQUATION: F(X+F(Y+Z)) = F(F(X+Y)+Z)) AND RB = B[0, 1) IS ITS LARGEST SUBGROUP. AS A GENERALIZATION OF THIS TOPIC, DECOMPOSER AND ASSOCIATIVE FUNCTIONS ON BINARY SYSTEMS (MAGMAS), SEMIGROUPS AND GROUPS ARE INTRODUCED AND STUDIED IN [2]. IF (X, ·) IS A BINARY SYSTEM AND F AN ARBITRARY FUNCTION FROM X TO X, THEN ANOTHER BINARY OPERATION (IN X) IS DEFINED BY X ·F Y = F(XY). THE BINARY SYSTEM (X, ·F) IS A SEMIGROUP IF AND ONLY IF F IS ASSOCIATIVE IN (X, ·). IN [2] WE SOLVED ASSOCIATIVE EQUATIONS IN ARBITRARY GROUPS AND PROVED THAT THE ASSOCIATIVE EQUATION DOES NOT HAVE ANY NONTRIVIAL SOLUTIONS IN THE SIMPLE GROUPS. IN THIS WAY ALL ASSOCIATIVE BINARY OPERATIONS ·F FOR A GROUP (G, ·) ARE CHARACTERIZED.IN THIS TALK I DISCUSS ABOUT THE F-MULTIPLICATION AND THE SEMIGROUP (G, ·F), WHERE F: G®G IS ASSOCIATIVE, AND SHOW THAT (F(G), ·F) IS A GROUP, NAMELY F-DECOMPOSITIONAL GROUP, AND IT IS THE LARGEST SUBGROUP OF THE SEMIGROUP (G, ·F ). ALSO SOME OTHER PROPERTIES OF F-DECOMPOSITIONAL GROUPS WILL BE CONSIDERED.

SOME EQUIVALENT CONDITIONS FOR A PSEUDO BCK-ALGEBRA TO BE COMMUTATIVE IS INVESTIGATED AND THAT WHEN A BOUNDED COMMUTATIVE PSEUDO BCK-ALGEBRA CAN BE A LATTICE IS VERIFIED. FINALLY, IT IS SHOWN THAT ANY PSEUDO BCK-ALGEBRA, UNDER SUITABLE CONDITIONS, IS A HEYTING.

WE STUDY SOME BASIC PROPERTY OF CLEANNESS. WE SHOW THAT IF R IS A NOTHERIAN RING AND M IS AN ALMOST CLEAN R-MODULE WITH THE PROPERTY THAT R/P IS COHEN–MACAULAY FOR ANY PÎASS(M), THEN DEPTH (M)=MIN{DIM(R/P) : PÎASS(M)}. USING THIS FACT SHOW THAT IF M IS A CLEAN R-MODULE AND ALL MINIMAL PRIME IDEALS OF M ARE COHEN–MACAULAY AND HAVE EQUAL HEIGHT, THEN M IS COHEN–MACAULAY. WE ALSO DISCUSS THE RELATION BETWEEN CLEANNESS AND SHALLEBLITY AND GIVE A SIMPLE PROOF FOR A THEOREM OF DRESS. FINALLY WE GIVE AN EASY PROOF FOR THE WELL KNOWN FACT THAT A PURE SHELLABLE SIMPLICIAL COMPLEX IS COHEN–MACAULAY.

IN THIS PAPER WE CONSTRUCT ALL LOCALLY EXTENDED AFFINE LIE ALGEBRAS IN GENERAL. WE DESCRIBE A CONSTRUCTION, WHICH STARTS WITH ANY CENTERLESS (Q(D)L)−GRADED LIE ALGEBRA L WITH D A LOCALLY IRREDUCIBLE FINITE ROOT SYSTEM AND L A TORSION- FREE ABELIAN GROUP, AND PRODUCE A LOCALLY EXTENDED AFFINE LIE ALGEBRA L WITH CENTERLESS CORE L.

IN THIS PAPER THE NOTIONS OF PRIME AND MAXIMAL SUBNEXUSES OF A NEXUS ARE DEFINED AND THE RELATIONSHIP BETWEEN OF THEM IS INVESTIGATED. IN PARTICULAR, ALL PRIME AND MAXIMAL SUBNEXUSES OF A NEXUS ARE CHARACTERIZED.

IN THIS TALK, THE NOTION OF INJECTIVITY IN THE CATEGORY POS-S OF S-POSETS, FOR A POMONOID S, IS DISCUSSED. FIRST WE SEE THAT, ALTHOUGH THERE IS NO NON-TRIVIAL INJECTIVE S-POSET WITH RESPECT TO MONOMORPHISMS, POS-S HAS ENOUGH (REGULAR) INJECTIVES WITH RESPECT TO REGULAR MONOMORPHISMS (SUB S-POSETS). THEN, RECALLING BANASCHEWSKI’S THEOREM WHICH STATES THAT REGULAR INJECTIVITY OF POSETS WITH RESPECT TO ORDER-EMBEDDINGS AND COMPLETENESS ARE EQUIVALENT, WE STUDY IT FOR S-POSETS AND GET SOME HOMOLOGICAL CLASSIFICATION OF POMONOIDS AND POGROUPS. AMONG OTHER THINGS, WE ALSO SEE THAT REGULAR INJECTIVE S-POSETS ARE EXACTLY THE RETRACTS OF COFREE S-POSETS OVER COMPLETE POSETS.

LET R BE A COMMUTATIVE RING AND LET I BE AN IDEAL OF R. LET M BE A NOETHERIAN R-MODULES. LET F BE AN R-MODULE WHICH IS FLAT RELATIVE TO M. SUPPOSE M0 IS A SUBMODULE OF M AND LET T = FÄR −.THEN IT IS SHOWN THAT THE SEQUENCES OF SETS &NBSP;ASSR(T(M)/IN(T(M’))) AND ASSR(IN(T(M))/IN(T(M’))), NÎN ARE ULTIMATELY CONSTANT.

WE STATE A NECESSARY AND SUFFICIENT CONDITION FOR EQUALITY OF TWO NONZERO DECOMPOSABLE SYMMETRIZED TENSORS, AND ALSO A NECESSARY AND SUFFICIENT CONDITION FOR ONE DECOMPOSABLE SYMMETRIZED TENSOR TO BE ZERO, WHEN THE SYMMETRIZER IS ASSOCIATED WITH AN IRREDUCIBLE CHARACTER OF A YOUNG SUBGROUP OF THE SYMMETRIC GROUP OF DEGREE M.

FOR A COMPLEX FINITE DIMENSIONAL SIMPLE LIE ALGEBRA G AND A FIELD K, ONE CAN DEFINE A LIE ALGEBRA G (K):= G Z Ä ZK OVER K WHERE G Z IS THE CHEVALLEY Z-FORM OF G WITH RESPECT TO A GIVEN CHEVALLEY BASIS OF G.

LET G BE A GROUP AND LET AUTF(G) DENOTE THE GROUP OF ALL AUTOMORPHISMS OF G CENTRALIZING G/F(G) ELEMENT WISE. IN THIS PAPER, WE FIRST CHARACTERIZE THE FINITE P-GROUPS G WITH CYCLIC FRATTINI SUBGROUP FOR WHICH |AUTF(G):INN(G)|=P. WE THEN GIVE A NECESSARY AND SUFFICIENT CONDITION ON A FINITE P-GROUP G FOR THE GROUP AUTF(G) TO BE ELEMENTARY ABELIAN.

LET D BE AN F-CENTRAL DIVISION ALGEBRA OF DEGREE N. IT IS PROVED THAT IF EITHER N=PA, P A PRIME, AND D* HAS A NON-ABELIAN NILPOTENT SUBGROUP OR N = PQ, WHERE P AND Q ARE PRIMES WITH P<Q SUCH THAT P DOES NOT DIVIDE Q-1 AND D* CONTAINS AN IRREDUCIBLE NON-ABELIAN SOLUBLE-BY-FINITE SUBGROUP, THEN G1(D)¹1.

LET M BE A UNITARY MODULE OVER A COMMUTATIVE RING R. WE SAY THAT M HAS FMP PROPERTY IF EVERY PRIME SUBMODULE OF M CONTAINS A FINITELY MANY PRIME SUBMODULES OF M. IN [4] WE HAVE GIVEN A CHARACTERIZATION OF FREE MODULES WITH THIS PROPERTY. THE MAIN AIM OF THIS NOTE IS TO STUDY SOME PROPERTIES OF RINGS AND MODULES WITH FMP PROPERTY. ALSO WE WILL GENERALIZE COHEN’S THEOREM FOR THE NOETHERIAN RINGS TO THE MODULES WITH FMP PROPERTY.

LET M BE A CLASS OF (MONO)MORPHISMS IN A CATEGORY A. TO STUDY MATHEMATICAL NOTIONS, SUCH AS INJECTIVITY, TENSOR PRODUCTS, FLATNESS, ONE NEEDS TO HAVE SOME CATEGORICAL AND ALGEBRAIC INFORMATION ABOUT THE PAIR (A,M).IN THIS PAPER WE TAKE A TO BE THE CATEGORY ACT-S OF S-ACTS, FOR A SEMIGROUP S, AND MD TO BE THE CLASS OF SEQUENTIALLY DENSE MONOMORPHISMS (OF INTERESTS TO COMPUTER SCIENTISTS, TOO) AND STUDY THE CATEGORICAL PROPERTIES, SUCH AS LIMITS AND COLIMITS, OF THE PAIR (A,M). INJECTIVITY WITH RESPECT TO THIS CLASS OF MONOMORPHISMS HAVE BEEN STUDIED BY GIULI, EBRAHIMI, AND THE AUTHORS AND GOT INFORMATION ABOUT INJECTIVITY RELATIVE TO MONOMORPHISMS.

THE NOTION OF OBSTINATE PSEUDO-BCK IDEALS IS INTRODUCED AND SOME RELATED PROPERTIES ARE INVESTIGATED. WE INVESTIGATE THE RELATIONS BETWEEN THESE PSEUDO IDEALS AND SEVERAL PSEUDO-BCK IDEALS. WE GIVE THE FOLLOWING CHARACTERIZATION THEOREM OF OBSTINATE IDEALS IN PSEUDO-BCK ALGEBRAS: LET I BE A PSEUDO IDEAL OF PSEUDO-BCK ALGEBRA X. THEN I IS OBSTINATE PSEUDO-BCK IDEAL IF AND ONLY IF I IS PSEUDO MAXIMAL, PSEUDO IRREDUCIBLE, PRIME, POSITIVE IMPLICATIVE OR IMPLICATIVE. IN THIS TALK WE DISCUSS ABOUT A RELATION BETWEEN OBSTINATE PSEUDO-BCK IDEALS AND OTHER ABOVE PSEUDO-BCK IDEALS.

ALTHOUGH THE BAER CRITERION FOR INJECTIVITY (IDEAL INJECTIVITY IMPLIES INJECTIVITY) IS TRUE FOR MODULES OVER A RING (WITH AN IDENTITY), IT IS AN OPEN PROBLEM FOR ACTS OVER A SEMIGROUP S (WITH OR WITHOUT IDENTITY). IN FACT, WE ARE NOT AWARE OF ANY TYPE OF WEAK INJECTIVITY IMPLYING INJECTIVITY OF S-ACTS, IN GENERAL, OTHER THAN SKORNJAKOV-BAER CRITERION, WHICH SAYS THAT INJECTIVITY WITH RESPECT TO SUBACTS OF CYCLIC ACTS IMPLIES INJECTIVITY WITH RESPECT TO ALL MONOMORPHISMS.

IN THIS TALK, BY CONSIDERING THE NOTION OF MV-ALGEBRA, WE ARE CONCERNED WITH A RELATIONSHIP BETWEEN ROUGH SET AND MV-ALGEBRA THEORY. WE SHALL INTRODUCE THE NOTION OF ROUGH SUBALGEBRA (RESP. IDEAL) WITH RESPECT TO AN IDEAL OF AN MV-ALGEBRA, WHICH IS AN EXTENDED NOTION OF SUBALGEBRA (RESP. IDEAL) IN AN MV-ALGEBRA. ALSO WE SHALL GIVE SOME PROPERTIES OF THE LOWER AND THE UPPER APPROXIMATIONS IN AN MV-ALGEBRA.

LET R BE A RING. IF WE REPLACE THE ORIGINAL ASSOCIATIVE PRODUCT OF R WITH THEIR CANONIC LIE PRODUCT, OR [A,B]=AB−BA FOR EVERY A,B IN R, THEN R WOULD BE A LIE RING. WITH THIS NEW PRODUCT THE ADDITIVE COMMUTATOR SUBGROUP OF R OR [R,R] IS A LIE SUBRING OF R. HERSTEIN HAS SHOWN THAT IN A SIMPLE RING R WITH CHARACTERISTIC UNEQUAL TO 2, ANY LIE IDEAL OF R EITHER IS CONTAINED IN Z(R), THE CENTER OF R, OR CONTAINS [R,R]. HE ALSO SHOWED THAT IN THIS SITUATION THE LIE RING [R,R]/Z[R,R] IS SIMPLE. HERE WE GIVE AN ALTERNATIVE MATRIX PROOF FOR THESE RESULTS. AS WE SHOWED IT SEEMS THAT THE CHARACTERISTIC CONDITION CAN BE PUT ON A SMALLER SET OF SIMPLE RINGS.

IN 1958 C. C. CHANG DEVELOPED AN ALGEBRAIC VERSION OF LUKASIEWICZ PROPOSITIONAL LOGIC AND PROVIDED AN ALGEBRAIC PROOF OF THE COMPLETENESS. THE RESULTING ALGEBRAIC SYSTEM BECAME KNOWN AS AN MV-ALGEBRA. IN 1998 PETER H´AJEK INTRODUCED THE VARIETY OF BL-ALGEBRAS AND SHOWED THAT THE VARIETY OF MV-ALGEBRAS ACTUALLY IS A SUBVARIETY OF THE VARIETY OF BL-ALGEBRAS.IN OTHER WORDS, ANY MV-ALGEBRA CAN BE EASILY VIEWED AS A SPECIAL BL-ALGEBRA. LET A BE AN MV-ALGEBRA. WE DENOTE BY SPEC (A) THE SET OF PRIME IDEALS OF A. SPEC (A) CAN BE ENDOWED WITH A SPECTRAL TOPOLOGY IN THE SAME MANNER AS IN THE CASE OF A COMMUTATIVE RING OR A BOUNDED DISTRIBUTIVE LATTICE. HOWEVER THE ABSENCE OF A DISTRIBUTIVE LAW REQUIERS A REFORMULATION OF THE RESULTS KNOWN IN RING THEORY. THUS IT MAKES SENSE TO GENERALIZE THE NOTION OF PRIME SPECTRUM TO BL-ALGEBRAS.

IN THIS TALK SOME CHARACTERIZATIONS OF D-SUPPLEMENTED AND D-LIFTING MODULES ARE GIVEN AND ARE INVESTIGATED SOME PROPERTIES OF THESE MODULES.

IN THIS PAPER WE DEFINED THE NOTIONS OF FUZZY IMPLICATIVE FILTER, FUZZY POSITIVE IMPLICATIVE FILTER AND FUZZY FANTASTIC FILTER AND WE FIND SOME RELATIONS AMONG THEM. SPECIALLY, WE FIND THE RELATIONS BETWEEN THE FUZZY IMPLICATIVE FILTERS AND GODEL ALGEBRAS.

BLOCKING SET IN PG (N, Q) IS A SET OF POINTS THAT HAS NONEMPTY INTERSECTION WITH EVERY HYPERPLANE OF PG(N, Q). A BLOCKING SET THAT CONTAINS A LINE IS CALLED TRIVIAL. A BLOCKING SET IS CALLED MINIMAL IF NONE OF ITS PROPER SUBSETS ARE BLOCKING SETS. A COVER C FOR A GROUP G IS CALLED A ŸN-COVER WHENEVER C IS AN IRREDUNDANT MAXIMAL CORE-FREE N-COVER FOR G AND IN THIS CASE WE SAY THAT G IS A ŸN-GROUP.IN THIS PAPER WE GIVE RELATIONS BETWEEN BLOCKING SETS AND &NBSP;ŸN-GROUPS.

LET G BE A FINITE GROUP AND P(G) BE THE SET OF PRIME DIVISORS OF THE ORDER OF G. FOR TÎP(G) DENOTE BY NT(G) THE ORDER OF A NORMALIZE OF T- SYLOW SUBGROUP OF G AND PUT N(G)={NT(G) | TÎP(G)}. IN THIS TALK, WE DISCUSS ABOUT AN ANSWER TO THE FOLLOWING PROBLEM FOR THE SIMPLE GROUPS OF LIE TYPE BN, CN: LET L BE A FINITE NON-ABELIAN SIMPLE GROUP AND G BE A FINITE GROUP WITH N(L)=N(G). IS IT TRUE THAT L@G?ALSO, WE DISCUSS ABOUT DIFFERENCE BETWEEN ORDERS OF THE SOLVABLE SUBGROUPS OF THE NON-ISOMORPHIC SIMPLE GROUPS BN(Q) AND CN(Q).

IN THIS PAPER THE NOTION OF HW-DIVISIBLE MODULES AS A GENERALIZATION OF H-DIVISIBLE MODULES IS DEFINED. WE SHOW THAT SOME OF THE MOST IMPORTANT PROPERTIES OF H-DIVISIBLE MODULES ARE HOLD FOR HW-DIVISIBLE MODULES.