A COMMUTATIVE RING R HAS PROPERTY (A) IF EVERY FINITELY GENERATED IDEAL OF R CONSISTING ENTIRELY OF ZERO DIVISORS HAS A NONZERO ANNIHILATOR. WE CONTINUE IN THIS PAPER THE STUDY OF RINGS WITH PROPERTY (A). WE EXTEND PROPERTY (A) TO R [X, A, D ], AND STUDY SUCH RINGS. THEN WE SHOW IF R IS REVERSIBLE AND  F: M®AUT (R) A COMPATIBLE MONOID HOMOMORPHISM, THEN R*M IS MCCOY. CONSEQUENTLY SEVERAL KNOWN RESULTS ON RIGHT MCCOY RINGS ARE EXTENDED TO A GENERAL SETTING.

FOR A MONOID M, WE INTRODUCE J-M-ARMENDARIZ RINGS, WHICH IS A COMMON GENERALIZATION OF J-ARMENDARIZ AND WEAK M-ARMENDARIZ RINGS, AND INVESTIGATE THEIR PROPERTIES.WE SHOW THAT EVERY NI-RING IS J-M-ARMENDARIZ, FOR ANY UNIQUE PRODUCT MONOID M. ALSO, WE PROVIDE VARIOUS EXAMPLES AND CLASSIFY HOW THE J-M-ARMENDARIZ RINGS BEHAVE UNDER VARIOUS RING EXTENSIONS. IT IS SHOWN THAT IF R IS SEMICOMMUTATIVE RING AND M-ARMENDARIZ THEN R IS J- (MXN) -ARMENDARIZ, WHERE N IS A UNIQUE PRODUCT MONOID.

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.

IN THIS PAPER, WE INTRODUCE B3 NEAR - RINGS AND GIVE SOME EXAMPLES. BY SOME EXAMPLES AND THEOREMS, WE FIND RELATIONS BETWEEN B3 NEAR - RINGS, B1 NEAR - RINGS AND STRONG B1 NEAR - RINGS. FINALLY, WE SHOW THAT EVERY B3 NEAR - RING N IS ISOMORPHIC TO A SUB DIRECT PRODUCT OF SUB DIRECTLY IRREDUCIBLE B3 NEAR - RINGS.

LET R BE A RING AND V BE A MATRIX VALUATION ON R, M (R) AND GL (R) WILL REPRESENT, AN ASSOCIATIVE RING WITH UNIT, THE SET OF ALL SQUARE MATRICES, AND THE SET OF INVERTIBLE MATRICES OVER R RESPECTIVELY.