Journal of Algebra 223, 477᎐488Ž. 2000 doi:10.1006rjabr.1999.8017, available online at http:rrwww.idealibrary.com on Armendariz Rings and Reduced Rings Nam Kyun Kim* Department of Mathematics, Kyung Hee Uni¨ersity, Seoul 130-701, Korea View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector and Yang Lee² Department of Mathematics Education, Pusan National Uni¨ersity, Pusan 609-735, Korea Communicated by Susan Montgomery Received February 18, 1999 Key Words: Armendariz ring; reduced ring; classical quotient ring. 1. INTRODUCTION Throughout this paper, all rings are associative with identity. Given a ring R, the polynomial ring over R is denoted by Rxwx. This paper concerns the relationships between Armendariz rings and reduced rings, being motivated by the results inwx 1, 2, 7 . The study of Armendariz rings, which is related to polynomial rings, was initiated by Armendarizwx 2 and Rege and Chhawchhariawx 7 . A ring R is called Armendariz if whenever s q q иии q m s q q иии q n polynomials fxŽ. a01ax axm , gxŽ. b01bx bxn g wx s s Rx satisfy fxgxŽ.Ž.0, then abij 0 for each i, j. Ž The converse is obviously true.. A ring is called reduced if it has no nonzero nilpotent elements. Reduced rings are Armendariz bywx 2, Lemma 1 and subrings of Armendariz rings are also Armendariz obviously. We emphasize the con- nections among Armendariz rings, reduced rings, and classical quotient rings. Moreover several examples and counterexamples are included for answers to questions that occur naturally in the process of this paper. *E-mail: [email protected]. ² E-mail: [email protected]. 477 0021-8693r00 $35.00 Copyright ᮊ 2000 by Academic Press All rights of reproduction in any form reserved. 478 KIM AND LEE 2. ARMENDARIZ RINGS First we consider some examples and counterexamples for Armendariz rings. Rege and Chhawchhariawx 7 showed that every n-by-n full matrix ring over any ring is not Armendariz, where n G 2. We have a similar result in the following. EXAMPLE 1. Let R be a ring. We claim that n-by-n upper triangular matrix rings over R are not Armendariz, where n G 2. It is enough to show that the 2-by-2 upper triangular matrix ring over R is not Armendariz because each subring of an Armendariz ring is also Armendariz. Let S be Ž.sqŽ.10 the 2-by-2 upper triangular matrix ring over R, and let fx 00 y Ž.1 10001Ž.sqŽ.Ž. wx Ž.Ž. 0 0 x and gx 01 01 x be polynomials in Sx. Then fxgx s Ž.Ž.10 01 / 0, but 00 01 0. So S is not Armendariz and consequently every n-by-n upper triangular matrix ring over R is not Armendariz. But we may find subrings of the 3-by-3 upper triangular matrix rings which may be Armendariz as in the following. PROPOSITION 2. Let R be a reduced ring. Then abc S sN0 ad a, b, c, d g R ½5ž/00a is an Armendariz ring. Proof. We employ the method in the proof ofwx 7, Proposition 2.5 . First notice that for abc111 abc 222 0 ad11, 0 ad 22g S 00a 00a 0012 we can denote their addition and multiplication by q s q q q q Ž.Ž.Ža1111, b , c , d a 222, b , c , d 2a 1a 21, b b 21, c c 21, d d 2. and Ž.a1111, b , c , daŽ. 222, b , c , d 2 s q q q q Ž.aa12, ab 12ba 12, ac 12bd 12ca 12, ad 12da 12, respectively. So every polynomial in Sxwxcan be expressed in the form wx ŽŽ.Ž.Ž.Ž..px0123, px, px, px for some pxi Ž.'s in Rx. ARMENDARIZ RINGS AND REDUCED RINGS 479 s s Let fxŽ. Žfx0123 Ž., fx Ž., fx Ž., fx Ž..and gx Ž. Žgx0 Ž., gx 1 Ž., gx 2 Ž., wx s s gx3Ž..be elements of Sx. Assume that fxgxŽ.Ž.0. Then fxgx Ž.Ž. q q q ŽŽ.Ž.fxgx00, fxgx 01 Ž.Ž.fxgx 10 Ž.Ž., fxgx 02 Ž.Ž.fxgx 13 Ž.Ž. q s fxgx20Ž. Ž., fxgx 03 Ž. Ž.fxgx 30 Ž. Ž.. 0. So we have the following system of equations: s Ž.0 fxgx00Ž. Ž. 0; q s Ž.1 fxgx01Ž. Ž.fxgx 10 Ž. Ž. 0; q q s Ž.2 fxgx02 Ž. Ž.fxgx 13 Ž. Ž.fxgx 20 Ž. Ž. 0; q s Ž.3 fxgx03Ž. Ž.fxgx 30 Ž. Ž. 0. wx s Use the fact that Rx is reduced. From Eq.Ž. 0 , we see that gxfx00 Ž. Ž. 0. q If we multiply Eq.Ž. 1 on the right side by fx0010Ž., then fxgxfx Ž. Ž. Ž. s s s fxgxfx100Ž. Ž. Ž.0. So fxgx01 Ž. Ž. 0 and hence fxgx10Ž. Ž. 0. Also if q we multiply Eq.Ž. 3 on the right side by fx0030Ž., then fxgxfx Ž. Ž. Ž. s s s fxgxfx300Ž. Ž. Ž.0. So fxgx03 Ž. Ž. 0 and hence fxgx30Ž. Ž. 0. Now if q we multiply Eq.Ž. 2 on the right side by fx0020Ž., then fxgxfx Ž. Ž. Ž. q s s fxgxfx130Ž. Ž. Ž.fxgxfx 200 Ž. Ž. Ž.0. So fxgx02 Ž. Ž. 0 and hence Eq. Ž.2 becomes Ј q s Ž.3 fxgx13Ž. Ž.fxgx 20 Ž. Ž. 0. Ј If we multiply Eq.Ž. 3 on the right side by fx11Ž., then we have fxgxŽ.3 Ž. s s 0 and so fxgx20Ž. Ž. 0. Now let X X X a b c nmabciii jjj XX s ijs fxŽ. ÝÝ0 adiix and gxŽ. 0 adjjx , X is0 00a js0 00a 0i 0j s ni s ni s ni s where fx0Ž. Ýis0 axi , fx1Ž. Ýis0 bxi , fx2Ž. Ýis0 cxi , fx3Ž. ni s m X j s m X j s m X j Ýis0 dxi , gx0Ž. Ý js0 axj , gx1Ž. Ý js0 bxj , gx2Ž. Ý js0 cxj , and s m X j XXXXs s s gx3Ž. Ý js0 dxjijijijij. Then we obtain that aa 0, ab 0, ba 0, ac s X s X s X s X s 0, bdij0, ca ij0, ad ij 0, and daij 0 for all i, j by the preced- ing results, the condition that R is reduced, andwx 2, Lemma 1 . Conse- quently X X X a b c abciii jjj XXs 0 adii0 adjj 0 X 00a 00a 0i 0j for all i, j and therefore S is an Armendariz ring. 480 KIM AND LEE Let S be a reduced ring and let ¡ иии ¦ aa12 a 13 a1n иии 0 aa23 a2 n sN~¥иии g Rnij00 a a3n a, a S . .. .. ¢§.. .. 000 0 иии a Based on Proposition 2, one may suspect that Rn may be also an Armen- dariz ring for n G 4. But the following example erases the possibility. EXAMPLE 3. Let S be a ring and let ¡ ¦ aa12 a 13 a 14 ~¥0 aa a R sN23 24 a, a g S . 4 00 aa ij ¢§34 000 0 a Let 0100 01y10 0000 00 00 fxŽ.sq x 0000 00 00 000000 00 00 and 0000 0000 0000 0001 gxŽ.sqx 0001 0001 000000 0000 wx s be polynomials in Rx4 . Then fxgxŽ.Ž. 0, but 0100 0000 0000 0001 / 0. 0000 0001 000000 0000 G So R4 is not Armendariz. Similarly, for the case of n 5, we have the same result. ¨ Given a ring R and a bimodule RRM , the tri ial extension of R by M is the ring TRŽ., M s R [ M with the usual addition and the multiplication s q Ž.Ž.Žr112, mr, m 2rr 1212, rm mr 12.. ARMENDARIZ RINGS AND REDUCED RINGS 481 Ž.rm g This is isomorphic to the ring of all matrices0 r , where r R and m g M and the usual matrix operations are used. COROLLARY 4wx 7, Proposition 2.5 . Let R be a reduced ring. Then the tri¨ial extension TŽ. R, R is an Armendariz ring. Proof. Notice that TRŽ., R is isomorphic to ab0 U sN0 a 0 a, b g R ½5ž/00a and that each subring of an Armendariz ring is also Armendariz. Thus TRŽ., R is an Armendariz ring by Proposition 2. From Corollary 4, one may suspect that if R is Armendariz then the trivial extension TRŽ., R is Armendariz. But the following example elimi- nates the possibility. sNÄ4Ž.ab g EXAMPLE 5. Let T be a reduced ring. Then R 0 a a, b T is an sNÄ4Ž.AB g Armendariz ring by Corollary 4. Let S 0 A A, B R and let 01 00 01 y10 ž/ž/ž/ž00 00 00 0y1 / fxŽ.sq x 00 01 00 01 00ž/ž/ž/ž/00 00 00 00 and 01 00 01 10 ž/ž/ž/ž/00 00 00 01 gxŽ.sqx 00 01 00 01 00ž/ž/ž/ž/00 00 00 00 be polynomials in Sxwx. Then fxgxŽ.Ž.s 0, but 01 00 01 10 ž/ž/ž/ž/00 00 00 01 / 0. 00 01 00 01 00ž/ž/ž/ž/00 00 00 00 Thus S is not Armendariz. By Anderson and Camillowx 1, Theorem 2 , polynomial rings over Armen- dariz rings are also Armendariz. So we may conjecture that skew polyno- 482 KIM AND LEE mial rings over Armendariz rings are also Armendariz.