The Armendariz Property Of Quotient Rings Of Polynomial Rings On Division Rings

In 1974,Armendariz discovered and proved that a reduced ring(a ring without nonzero nilpotent elements)satisfies the following condition:For two polynomials f(x)=(?)aixi and g(x)=(?)bjxj over R,if f(x)g(x)=0,then aibj=0(0?i?m,0?j?n).In 1997,Rege and Chhawchharia named a ring R satisfying the above condition an Armendariz ring.In recent years,Armendariz ring theory has also been developed rapidly,and a large number of research results have emerged continuously.A quotient ring of an Armendariz ring is not necessarily an Armendariz ring.Therefore,it is worth discussing which quotient ring of an Armendariz ring has Armendariz property for polynomial rings.In this paper,we give some conditions under which the quotient ring of polynomial ring D[x]modular binomial or trinomial on division ring D is an Armendariz ring,and the necessary and sufficient condition under which the quotient ring of polynomial ring H[x]modular trinomial on the quaternion division ring H is an Armendariz ring.The main results are as follows.Let Z be the center of D,Z(q)={q'?D|qq'?q'q} and D2={q2|q?D}.Theorem 2.15.Let D be a division ring.Then D[x]/(X2-q)is an Armendariz ring if and only if one of the following conditions holds:1.q(?)Z;2.q?Z2;3.q?Z\D2;4.charD=2,q?Z?D2.Theorem 2.19 Let D be a division ring and n>m>0.If Z(b)? Z(c),then D[x]/(xn+bxm+c)is an Armendariz ring.Theorem 2.20.Let D be a division ring and charD?2.Then D[x]/(x2+bx+c)is an Armendariz ring if and only if one of the following conditions holds:1.b,c?Z and b2-4c?Z2?(Z\D2);2.b(?)Z or c(?)Z.Theorem 2.21.Let D be a division ring of characteristic 0.If xn-1 is a product of polynomials of degree 1,then D[x]/(xn-1)is an Armendariz ring if and only if xn-1 has n roots in Z(counting multiplicities).Theorem 3.7 Let H be the quaternion algebra over the real number field R and n>m>0.Then H[x]/(xn+axm+?)is an Armendariz ring if and only if one of the following conditions holds:1.?,??R and f(x)=xn+axm+? has n roots in R;2.??R,?(?)R;3.?(?)R,??R;4.?,?(?)R and ?????;5.?,?(?)R,? and ? are R-linearly dependent;6.?,?(?)R,??=??,and ?,? are R-linearly independent,and one of the following conditions holds:(a)? is not a root of xn/d+axm/d+?;(b)d=2,?>0 is a root of xn/d+axm/d+?(c)d=1,? is a root of xn+axm+?.where d is the greatest common divisor of m and n and d=pm+qn,?=(-1)p+q(?-?t)qtp is a non-zero real number and t is a real number such that(?-Re?)?(?-Re?)t.