Extensions of Reduced Rings

WU Hui-feng

(College of Science,Hangzhou Normal University,Hangzhou 310036,China)

Extensions of Reduced Rings

WU Hui-feng

(College of Science,Hangzhou Normal University,Hangzhou 310036,China)

A ringRis a reduced ring,provided thata2=0 implies thata=0.The paper discussed the relations between reduced rings and 3-Armendariz rings and proved that power series rings and some special upper triangular matrix rings of reduced rings are 3-Armendariz rings.

reduced ring; power series ring; 3-Armendariz ring.

1 Introduction

Condition(P) For alla,b,c∈R,if (abc)2=0,thenabc=0.(see [1])

Proposition1 IfRis a reduced ring,thenRsatisfies the Condition (P),but the converse is not true.

ProofIt is easy to prove thatRis a reduced ring implies thatRsatisfies the Condition (P),there exists a ring that satisfies the Condition (P) but is not a reduced ring.Let

From [1],we know thatRis 3-Armendariz ring if and only ifR[x] is 3-Armendariz ring.Clearly,all subrings of 3-Armendariz rings are 3-Armendariz rings.IfR[[x]] is a 3-Armendariz ring,thenR[x] is a 3-Armendariz ring,but the converse is not true.

Theorem1 LetRbe a reduced ring,thenR[[x]] is a 3-Armendariz ring.

If [f(x)g(x)h(x)]2=0,that is

(d0+d1x+d2x2+d3x3+…+dn-1xn-1+dnxn)·(d0+d1x+d2x2+d3x3+…+dn-1xn-1+dnxn)=

dn+1dn-1)x2n+(d0d2n+1+d2n+1d0+d1d2n+d2nd1+d2d2n-1+d2n-1d2+…+dndn+1+dn+1dn)x2n+1+…=0.

SetAibe the coefficient of [f(x)g(x)h(x)]2.then

d2n-2d2+…+dn-1dn+1=0;A2n+1=d0d2n+1+d2n+1d0+d1d2n+d2nd1+d2d2n-1+d2n-1d2+…+

dn-1dn+2+dndn+1=0; ….

…

AsA2n=0 andd0=0,d1=0,d2=0,d3=0,…,dn-1=0,

Continuing in this way we haved0=0,d1=0,d2=0,d3=0,…,dn=0,….

Corollary1 IfRis a reduced ring,thenR[x] is a 3-Armendariz ring.

Theorem2 LetRbe a reduced ring,then is a 3-Armendariz ring.

ProofIt is well know that for a ringRand any positive integern≥2,R[x]/(xn)≌S.where (xn) is the ideal ofR[x] generated byxn.It is evident thatR[x]/(xn)≌R′,R′ is subring ofR[[x]],soR′≌S.SinceRis reduced ring,by Theorem 1,we knowR[[x]] is 3-Armendariz ring,moveover,subrings of 3-Armendariz rings are 3-Armendariz rings,soR′ is a 3-Armendariz ring.ThereforeSis a 3-Armendariz ring and the proof is complete.

Theorem3 LetRbe a reduced ring,then

is a 3-Armendariz ring.

ProofSinceRis a reduced ring,thenRsatisfies the Condition (P),that is

if(abc)2=0,thenabc=0.

InR,since (bca)2=bcabca=bc(abc)a=0,sobca=0.

We can denote their addition and multiplication by:

(f0(0),f0(x),f1(x))+(g0(0),g0(x),g1(x))=(f0(0)+g0(0),f0(x)+g0(x),f1(x)+g1(x)).and

(f0(0),f0(x),f1(x))·(g0(0),g0(x),g1(x))=(f0(0)g0(0),f0(x)g0(x),f0(0)g1(x)+f1(x)g0(x)).

So every polynomial ofR[y] can be expressed by (f0(0),f0(y),f1(y)),wheref0(y),f1(y)∈R[x][y].For allf(y),g(y),h(y) ∈R〈x〉[y],and

f(y)=(f0(0),f0(y),f1(y)),
g(y)=(g0(0),g0(y),g1(y)),
h(y)=(h0(0),h0(y),h1(y)).

Iff(y)g(y)h(y)=0,we have the following system of equations:

f0(0)g0(0)h0(0)=0,

(1)

f0(y)g0(y)h0(y)=0,

(2)

f0(0)g0(0)h1(y)+f0(0)g1(y)h0(y)+f1(y)g0(y)h0(y)=0.

(3)

If we multiply (3) on the right side byf0(y),then

f0(0)g0(0)h1(y)f0(y)+f0(0)g1(y)h0(y)f0(y)=0

(3′)

(sincef0(y)g0(y)h0(y)=g0(y)h0(y)f0(y)=0.)

Also if we multiply (3′) on the right side byg0(y),then

f0(0)g0(0)h1(y)f0(y)g0(y)=0.

Thusf0(0)g0(0)h1(y)f0(0)g0(0)=0.So (f0(0)g0(0)h1(y))2=f0(0)g0(0)h1(y)f0(0)g0(0)h1(y)=0.SinceRa reduced ring,thenR[x] is a reduced ring,and thenR[x][y] is a reduced ring.Thereforef0(0)g0(0)h1(y)=0.Hencef0(0)g1(y)h0(y)f0(y)=0,sof0(0)g1(y)h0(y)f0(0)=0,it means that (f0(0)g1(y)h0(y))2=0,thenf0(0)g1(y)h0(y)=0.

And sof0(0)g0(0)h1(y)=f0(0)g1(y)h0(y)=f1(y)g0(y)h0(y)=0.

Write

and set

For all 0≤i≤r,0≤j≤s,0≤k≤t,we have

we knowR[x][y] is a reduced ring,soR[x][y] is a 3-Armendariz ring.Sincef0(0)g0(0)h0(0)=0,thenf1i(0)f2j(0)f3k(0)=0.Sincef0(y)g0(y)h0(y)=0,thenf1i(x)f2j(x)f3k(x)=0.Sincef0(0)g0(0)h1(y)=0,thenf1i(0)f2j(0)g3k(x)=0.Sincef0(0)g1(y)h0(y)=0,thenf1i(0)g2j(x)f3k(x)=0.Sincef1(y)g0(y)h0(y)=0,theng1i(x)f2j(x)f3k(x)=0.

Consequently

HenceR〈x〉 is a 3-Armendariz ring.

Example1Z2〈x〉 is a 3-Armendariz ring,henceZ2〈x〉 is a Armendariz ring whereZ2is the field with two elements.

ProofIn view of Theorem 3,Z2〈x〉 is a 3-Armendariz ring.ButZ2〈x〉 has an identity,and so it is a Armendariz ring.

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约化环的推广

伍惠凤

(杭州师范大学理学院，浙江 杭州 310036)

称环R是约化环，如果a2=0，那么a=0.讨论了约化环和3-Armendariz环之间的关系，证明了不带单位元的约化环上的幂级数环和某些特殊的上三角矩阵环是3-Armendariz 环.

约化环; 幂级数环; 3-Armendariz环.

date：2011-03-18

Biography：Wu Hui-feng(1982—),famale,born in Anqing,Anhui province,master,engageed in Algebraic.E-mail:yaya57278570@163.com

10.3969/j.issn.1674-232X.2011.05.005

O153.3MSC2010:16E99; 14F99ArticlecharacterA

1674-232X(2011)05-0407-04