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＝0implies 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］）

Proposition 1IfRis 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.

Theorem 1LetRbe a reduced ring，thenR［［x］］is a 3－Armendariz ring.

Corollary 1IfRis a reduced ring，thenR［x］is a 3－Armendariz ring.

Theorem 2LetRbe 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.

Theorem 3LetRbe a reduced ring，then

is a 3－Armendariz ring.

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

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

We can denote their addition and multiplication by：

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

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

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

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

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.

Example 1Z2〈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.

［1］Yang Suiyi.On the extension of Armendariz rings［D］.Lanzhou：Lanzhou University，2008：9－19.

［2］Anderson D D，Camillo V.Armendariz rings and Gaussian rings［J］.Comm Algebra，1998，26（7）：2265－2272.

［3］Rege M B.Chhawchharia S.Armendariz rings［J］.Proc Japan Acad Ser A Math Sci，1997，73：14－17.

［4］Hirano Y.On annihilator ideals of a polynomial ring over a non commutative ring［J］.J Pure Appl Algebra，2002，168：45－52.

［5］Yan Zhanping.Armendariz property of a class of matrix rings［J］.Journal of Northwest Normal University Natural Science，2003，39（3）：22－24.

［6］Wang Wenkang.Armendariz and semicommutative properties of a class of upper triangular matrix rings［J］.Journal of Shandong University：Natural science Edition，2008，43（2）：62－65.

［7］Kim N K，Lee K H，Lee Y，Power series rings satisfying a zero divisor porperty［J］.Comm Alg，2006，34：2205－2218.

約化環(huán)的推廣

伍惠鳳

（杭州師范大學理學院，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).

O153.3 MSC2010：16E99；14F99 Article character：A

1674－232X（2011）05－0407－04

10.3969/j.issn.1674－232X.2011.05.005

date：2011－03－18

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