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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約化環(huán)的推廣

伍惠鳳

(杭州師范大學(xué)理學(xué)院，浙江 杭州 310036)

稱環(huán)R是約化環(huán)，如果a2=0，那么a=0.討論了約化環(huán)和3-Armendariz環(huán)之間的關(guān)系，證明了不帶單位元的約化環(huán)上的冪級數(shù)環(huán)和某些特殊的上三角矩陣環(huán)是3-Armendariz 環(huán).

約化環(huán); 冪級數(shù)環(huán); 3-Armendariz環(huán).

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