Extended Armendariz Rings
aa r X i v : . [ m a t h . R A ] D ec EXTENDED ARMENDARIZ RINGS
NAZIM AGAYEV, ABDULLAH HARMANCI, AND SAIT HALICIOGLUA
BSTRACT . In this note we introduce central linear Armendariz rings as a generalizationof Armendariz rings and investigate their properties.
AMS Subject Classification:
Key words: reduced rings, central reduced rings, abelian rings, Armendariz rings, linearArmendariz rings, central linear Armendariz rings.
1. I
NTRODUCTION
Throughout this paper R denotes an associative ring with identity. Rege and Chhawch-haria [13], introduce the notion of an Armendariz ring. The ring R is called Armendariz iffor any f ( x ) = (cid:229) ni = a i x i , g ( x ) = (cid:229) sj = b j x j ∈ R [ x ] , f ( x ) g ( x ) = a i b j = i and j . The name of the ring was given due to Armendariz who proved that reduced rings(i.e. rings without nonzero nilpotent elements) satisfied this condition [2].Number of papers have been written on the Armendariz rings (see, e.g. [1], [9]). So far,Armendariz rings are generalized in different ways (see namely, [6], [12]). In particular,Lee and Wong [10] introduced weak Armendariz rings (i.e. if the product of two linearpolynomials in R [ X ] is 0, then each product of their coefficients is 0), Liu and Zhao [12]introduce also weak Armendariz rings ( if the product of two polynomials in R [ X ] is 0,then each product of their coefficients is nilpotent) as another generalization of Armen-dariz rings. To get rid of confusion, we call the rings linear Armendariz which satisfyLee and Wong condition. A ring R is called central linear Armendariz , if the productof two linear polynomials in R [ X ] is 0, then each product of their coefficients is central.Clearly, Armendariz rings are linear Armendariz and linear Armendariz rings are centrallinear Armendariz. In case R is reduced ring every weak Armendariz ring is central linearArmendariz. We supply some examples to show that the converses of these statementsneed not be true in general. We prove that the class of central linear Armendariz rings liesstrictly between classes of linear Armendariz rings and abelian rings. For a ring R , it isshown that the polynomial ring R [ x ] is central linear Armendariz if and only if the Laurent polynomial ring R [ x , x − ] is central linear Armendariz. Among others we also show that R is reduced ring if and only if the matrix ring T kn ( R ) is Armendariz ring if and only ifthe matrix ring T n − n ( R ) is central linear Armendariz ring, for a natural number n ≥ k = [ n / ] . And for an ideal I of R , if R / I central linear Armendariz and I is reduced, then R is central linear Armendariz.We also introduce central reduced rings as a generalization of reduced rings. The ring R is called central reduced if every nilpotent is central. We prove that if R is centralreduced ring, then R is central linear Armendariz, and if R is central reduced ring, then thetrivial extension T ( R , R ) is central linear Armendariz. Moreover, it is proven that if R isa semiprime ring, then R is central reduced ring if and only if R [ x ] / ( x n ) is central linearArmendariz, where n ≥ ( x n ) is the ideal generated by x n .We write R [ x ] , R [[ x ]] , R [ x , x − ] and R [[ x , x − ]] for the polynomial ring, the power se-ries ring, the Laurent polynomial ring and the Laurent power series ring over R , respec-tively. 2. C ENTRAL L INEAR A RMENDARIZ R INGS
In this section central linear Armendariz rings are introduced as a generalization oflinear Armendariz rings. We prove that some results of linear Armendariz rings can beextended to central linear Armendariz rings for this general settings. Clearly, every Ar-mendariz ring is linear Armendariz. However, linear Armendariz rings are not necessarilyArmendariz in general (see [10, Example 3.2 ]).We now give a possible generalization of linear Armendariz rings.
Definition 2.1.
The ring R is called central linear Armendariz if the product of two linearpolynomials in R [ X ] is , then each product of their coefficients is central. Note that all commutative rings, reduced rings, Armendariz rings and linear Armendarizrings are central linear Armendariz. It is clear that subrings of central linear Armendarizrings are central linear Armendariz.Recall that R is said to be abelian if idempotent elements of R are central. Lemma 2.2.
If the ring R is central linear Armendariz, then R is abelian.Proof.
Let e be any idempotent in R , consider f ( x ) = e − er ( − e ) x , g ( x ) = ( − e ) + er ( − e ) x ∈ R [ x ] for any r ∈ R . Then f ( x ) g ( x ) =
0. By hypothesis, in particular er ( − e ) is central. Therefore er ( − e ) =
0. Hence er = ere for all r ∈ R . Similarly we consider XTENDED ARMENDARIZ RINGS 3 h ( x ) = ( − e ) − ( − e ) rex and t ( x ) = e +( − e ) rex in R [ x ] for any r ∈ R . Then h ( x ) t ( x ) = ( − e ) re = ere = re for all r ∈ R . It follows that e is central element of R ,that is, R is abelian. (cid:3) Example 2.3.
Let R be any ring. For any integer n ≥ , consider the ring R n × n of n × nmatrices and the ring T n ( R ) of n × n upper triangular matrices over R. The rings R n × n and T n ( R ) contain non-central idempotents. Therefore they are not abelian. By Lemma 2.2these rings are not central linear Armendariz. Recall that a ring R is semicommutative , if for any a , b ∈ R , ab = aRb = Theorem 2.4.
Let R be a von Neumann regular ring R. Then the following are equivalent: ( ) R is Armendariz. ( ) R is reduced. ( ) R is central linear Armendariz. ( ) R is linear Armendariz. ( ) R is semicommutative.Proof.
By Lemma 2.2 and [5, Lemma 3.1, Theorem 3.2], we have ( ) ⇒ ( ) . ( ) ⇒ ( ) Clear. ( ) ⇒ ( ) Let a = a ∈ R . By ( ) , aRa =
0. So ( aR ) =
0. Assume aR = aR contains a non-zero idempotent. This is a contradiction. Hence a = (cid:3) We now give a condition for a ring to be central linear Armendariz relating to centralidempotents.
Lemma 2.5.
Let R be a ring and e an idempotent of R. If e is a central idempotent of R,then the following are equivalent: (1)
R is central linear Armendariz. (2) eR and ( − e ) R are central linear Armendariz.Proof. (1) ⇒ (2) Since the subrings of central linear Armendariz rings are central linearArmendariz, ( ) holds.(2) ⇒ (1) Let f ( x ) = a + a x , g ( x ) = b + b x be non zero polynomials in R [ x ] . Assumethat f ( x ) g ( x ) =
0. Let f = e f ( x ) , f = ( − e ) f ( x ) , g = eg ( x ) , g = ( − e ) g ( x ) . Then f ( x ) g ( x ) = ( eR )[ x ] and f ( x ) g ( x ) = (( − e ) R )[ x ] . By (2) ea i eb j is central in eR and ( − e ) a i ( − e ) b j is central in ( − e ) R for all 0 ≤ i ≤
1, 0 ≤ j ≤
1. Since e and NAZIM AGAYEV, ABDULLAH HARMANCI, AND SAIT HALICIOGLU − e central in R , R = eR ⊕ ( − e ) R and so a i b j = ea i b j + ( − e ) a i b j is central in R forall 0 ≤ i ≤
1, 0 ≤ j ≤
1. Then R is central linear Armendariz. (cid:3) Clearly, any linear Armendariz ring is central linear Armendariz. We now prove thatthe converse is true if the ring is right p . p . − ring. Theorem 2.6.
If the ring R is linear Armendariz, then R is central linear Armendariz. Theconverse holds if R is right p . p . − ring.Proof. Suppose R is central linear Armendariz and right p . p . − ring. Let f ( x ) = a + a x , g ( x ) = b + b x ∈ R [ x ] . Assume f ( x ) g ( x ) = a b = ( ) a b + a b = ( ) a b = ( ) By hypothesis there exist idempotents e i ∈ R such that r ( a i ) = e i R for all i . So b = e b and a e =
0. Multiply (2) from the right by e , by Lemma 2.2, R is abelian and wehave 0 = a b e + a b e = a e b + a b e = a b . So a b =
0. Hence R is linearArmendariz. This completes the proof. (cid:3) Let R be a ring and let M be an ( R , R ) -bimodule. The trivial extension of R by M isdefined to be the ring T ( R , M ) = R ⊕ M with the usual addition and the multiplication ( r , m )( r , m ) = ( r r , r m + m r ) .Example 2.7 shows that the assumption ”right p.p.-ring” in Theorem 2.6 is not super-fluous. Example 2.7.
There exists a central linear Armendariz ring which is neither right p.p.-ringnor linear Armendariz ring.Proof.
Let n be an integer with n ≥
2. Consider the ring R = T ( Z n , Z n ) . If a = n − and f ( x ) = ¯ a ¯0¯0 ¯ a + ¯ a ¯1¯0 ¯ a x ∈ R [ x ] , then ( f ( x )) =
0. Because ¯ a ¯0¯0 ¯ a ¯ a ¯1¯0 ¯ a = R is not a linear Armendariz ring. Since R is commutative, it is central linear Armendarizring. Moreover, since the principal ideal I = Z n = R is not projective, R is not right p.p.-ring. (cid:3) Now we will introduce a notation for some subrings of T n ( R ) . Let k be a natural numbersmaller than n . Say XTENDED ARMENDARIZ RINGS 5 T kn ( R ) = ( n (cid:229) i = j k (cid:229) j = a j e ( i − j + ) i + n − k (cid:229) i = j n − k (cid:229) j = r i j e j ( k + i ) : a j , r i j ∈ R ) where e i j ’ s are matrix units. Elements of T kn ( R ) are in the form x x ... x k a ( k + ) a ( k + ) ... a n x ... x k − x k a ( k + ) ... a n x ... a n ... x where x i , a js ∈ R , ≤ i ≤ k , ≤ j ≤ n − k and k + ≤ s ≤ n .For a reduced ring R , our aim is to investigate necessary and sufficent conditions for S = T kn ( R ) to be central linear Armendariz. In [11], Lee and Zhou prove that, if R isreduced ring, then S is Armendariz ring for k = [ n / ] . Hence S is linear Armendariz and so S is central linear Armendariz. In the following, we show that the converse of this theoremis also true. Moreover, it is proven that R is reduced ring if and only if T kn ( R ) is Armendarizring if and only if T n − n ( R ) is central linear Armendariz ring. In this direction, we need thefollowing lemma: Lemma 2.8.
Suppose that there exist a , b ∈ R such that a = b = and ab = ba is notcentral. Then R is not a central linear Armendariz ring.Proof. ( a + bx )( a − bx ) = R [ x ] , but ab is not central. So, R is not a central linearArmendariz ring. (cid:3) Theorem 2.9.
Let n ≥ be a natural number. Then R is reduced ring if and only if T kn ( R ) is central linear Armendariz ring, where ≤ k ≤ n − .Proof. Let R be a reduced ring. In [11], it is shown that T kn ( R ) is Armendariz ring and soit is central linear Armendariz. Conversely, suppose that R is not a reduced ring. Choose anonzero element a ∈ R with square zero. Then for elements A = a ( e + e + ... + e nn ) , B = e ( k + ) + e ( k + ) + ... + e n in T kn ( R ) , A = B = AB = BA is not central, since ( AB )( e ( n − k ) + e ( n − k + ) + ... + e k ( n − ) + e ( k + ) n ) = ae n =
0. Therefore, from Lemma2.8, T kn ( R ) is not central linear Armendariz ring. This completes the proof. (cid:3) Theorem 2.10.
Let R be a ring, n ≥ be a natural number and k = [ n / ] . Then thefollowing are equivalent: NAZIM AGAYEV, ABDULLAH HARMANCI, AND SAIT HALICIOGLU ( ) R is reduced ring. ( ) T kn ( R ) is Armendariz ring. ( ) T n − n ( R ) is central linear Armendariz ring.Proof. ( ) ⇒ ( ) See [11]. ( ) ⇒ ( ) Since subrings of Armendariz rings are Armendariz, the rest is clear. ( ) ⇒ ( ) It follows from Theorem 2.9. (cid:3)
Note that the homomorphic image of a central linear Armendariz ring need not be cen-tral linear Armendariz. If R is commutative and Gaussian ring, by [1, Theorem 8] everyhomomorphic image of R is Armendariz and so it is central linear Armendariz.In [7], it was shown that for a ring R , if I is a reduced ideal of R such that R / I isArmendariz, then R is Armendariz. For central linear Armendariz rings we have the similarresult. Theorem 2.11.
Let R / I be central linear Armendariz and I be reduced. Then R is centrallinear Armendariz.Proof.
Let a , b ∈ R . If ab =
0, then ( bIa ) =
0. Since bIa ⊆ I and I is reduced, bIa = ( aIb ) ⊆ ( aIb )( I )( aIb ) =
0. Therefore aIb =
0. Assume f ( x ) = a + a x , g ( x ) = b + b x ∈ R [ x ] and f ( x ) g ( x ) =
0. Then a b = ( ) a b + a b = ( ) a b = ( ) We first show that for any a i b j , a i Ib j = b j Ia i =
0. Multiply ( ) from the right by Ib ,we have a b Ib =
0, since a b Ib =
0. Then ( b Ia ) ⊆ b I ( a b Ia b ) Ia =
0. Hence b Ia =
0. This implies a Ib =
0. Multiply ( ) from the left by a I , we have a Ia b + a Ia b = a Ia b =
0. Thus ( b Ia ) = b Ia =
0. Therefore a Ib = R / I is central Armendariz, it follows that a i b j is central in R / I . So a i b j r − ra i b j ∈ I for any r ∈ R . Now from above results, it can be easily seen that ( a i b j r − ra i b j ) I ( a i b j r − ra i b j ) =
0. Then a i b j r = ra i b j for all r ∈ R . Hence a i b j is central for all i and j . Thiscompletes the proof. (cid:3) Let S denote a multiplicatively closed subset of R consisting of central regular elements.Let S − R be the localization of R at S . Then we have: XTENDED ARMENDARIZ RINGS 7
Proposition 2.12.
R is central linear Armendariz if and only if S − R is central linearArmendariz.Proof.
Suppose that R is a central linear Armendariz ring. Let f ( x ) = (cid:229) i = ( a i / s i ) x i , g ( x ) = (cid:229) j = ( b j / t j ) x j ∈ ( S − R )[ x ] and f ( x ) g ( x ) =
0. Then we may find u , v , c i and d j in S such that u f ( x ) = (cid:229) i = a i c i x i ∈ R [ x ] , vg ( x ) = (cid:229) i = b j d j x j ∈ R [ x ] and ( u f ( x ))( vg ( x )) =
0. By supposition ( a i c i )( b j d j ) are central in R for all i and j . Since c i and d j are regular central elements of R , a i b j are central in R for all i and j . It follows that ( a i / s i )( b j / t j ) are central for all i and j .Conversely, assume that S − R is a central linear Armendariz ring. Let f ( x ) = (cid:229) i = a i x i , g ( x ) = (cid:229) j = b j x j ∈ R [ x ] . Assume f ( x ) g ( x ) =
0. Then f ( x ) / = (cid:229) i = ( a i / ) x i , g ( x ) = (cid:229) j = ( b j / ) x j ∈ S − R [ x ] and ( f ( x ) / )( g ( x ) / ) = S − R . By assumption ( a i / )( b j / ) is central in S − R . Hence, for all i and j , a i b j is central in R . (cid:3) Corollary 2.13.
For any ring R, the polynomial ring R [ x ] is central linear Armendariz ifand only if the Laurent polynomial ring R [ x , x − ] is central linear Armendariz.Proof. Let S = { , x , x , x , x , ... } . Then S is a multiplicatively closed subset of R [ x ] con-sisting of central regular elements. Then the proof follows from Proposition 2.12. (cid:3) We now define central reduced rings as a generalization of reduced rings.
Definition 2.14.
The ring R is called central reduced ring if every nilpotent element iscentral.
Example 2.15.
All commutative rings, all reduced rings and all strongly regular rings arecentral reduced.
One may suspect that central reduced rings are reduced. But the following exampleerases the possibility.
Example 2.16.
Let S be a commutative ring and R = S [ x ] / ( x ) . Then R is commutativering and so R is central reduced. If a = x + ( x ) ∈ R, then a = . Therefore R is not areduced ring. It is well known that if the ring R is reduced, then R is linear Armendariz. In our case,we have the following: NAZIM AGAYEV, ABDULLAH HARMANCI, AND SAIT HALICIOGLU
Theorem 2.17.
If R is central reduced ring, then R is central linear Armendariz.Proof.
Let f ( x ) = a + a x , g ( x ) = b + b x ∈ R [ x ] . Assume f ( x ) g ( x ) =
0. Then we have : a b = ( ) a b + a b = ( ) a b = ( ) Since ( b a ) = ( b a ) = b a , b a ∈ C ( R ) , where C ( R ) is the center of R .Multiply (2) from the right by a , we have a b a + a b a =
0. Thus a b a + b a a = a , we have a b a = ( a b a ) =
0, thatis, a b a ∈ C ( R ) . Hence ( a b ) = a b ∈ C ( R ) . Similarly it can be shown that a b ∈ C ( R ) . (cid:3) Note that if R is reduced ring, by [13, Proposition 2.5] trivial extension T ( R , R ) is Ar-mendariz and so it is linear Armendariz. For central reduced rings, we have Lemma 2.18.
If R is central reduced ring, then the trivial extension T ( R , R ) is centrallinear Armendariz. The converse holds if R is semiprime.Proof. Let f ( x ) = a b a + a b a x = f ( x ) f ( x ) f ( x ) , g ( x ) = c d c + c d c x = g ( x ) g ( x ) g ( x ) ∈ T ( R , R )[ x ] . If f ( x ) g ( x ) = f ( x ) g ( x ) = f ( x ) g ( x ) f ( x ) g ( x ) + f ( x ) g ( x ) f ( x ) g ( x ) = . Hence f ( x ) g ( x ) = f ( x ) g ( x ) + f ( x ) g ( x ) = . In this case, we have a c = ( ) a c + a c = ( ) a c = ( ) From ( ) and ( ) , a c , a c ∈ C ( R ) and so c a , c a ∈ C ( R ) . Multiply (2) from the rightby a , we have a c a + a c a =
0. Thus a c a + c a a =
0, so a c a = ( a c a ) =
0, that is, a c a ∈ C ( R ) . Hence ( a c ) = a c ∈ C ( R ) . Similarly itcan be shown that a c ∈ C ( R ) .Conversely, suppose R is semiprime and S = T ( R , R ) is central linear Armendariz. Let XTENDED ARMENDARIZ RINGS 9 a n = a ∈ R . Consider f ( x ) = a n − a n − + a n − a n − x , g ( x ) = a n − a n − + a n − − a n − x ∈ S [ x ] . Then f ( x ) g ( x ) =
0. Hence a n − ∈ C ( S ) and so a n − ∈ C ( R ) . Therefore ( a n − R ) = a n − =
0. Continuing in thisway, we have a = (cid:3) In [1, Theorem 5], Anderson and Camillo proved that for a ring R and n ≥ T n − n ( R ) is Armendariz if and only if R is reduced. Lee andWong [10, Theorem 3.1] also proved that T n − n ( R ) is linear Armendariz if and only if R isreduced. For central linear Armendariz rings, we have the following. Theorem 2.19.
Let R be a semiprime ring and n ≥ a natural number. R is central reducedring if and only if T n − n ( R ) is central linear Armendariz.Proof. Suppose R is central reduced ring. Let a = a ∈ R . Then a ∈ C ( R ) and so aRa =
0. Since R is semiprime, we have a =
0. Therefore R is reduced and T n − n ( R ) isArmendariz by [1, Theorem 5]. Hence T n − n ( R ) is linear Armendariz and by Theorem2.6, it is central linear Armendariz. Conversely, assume that T n − n ( R ) is central linearArmendariz. Using the similar technique as in the proof of Lemma 2.18, it can be shownthat R is central reduced. (cid:3) R EFERENCES[1] D.D. Anderson and V. Camillo,
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AZIM A GAYEV , Q
AFQAZ U NIVERSITY , D
EPARTMENT OF P EDAGOGY , B
AKU , A
ZERBAIJAN
E-mail address : [email protected] A BDULLAH H ARMANCI , H
ACETTEPE U NIVERSITY , D
EPARTMENT OF M ATHEMATICS , A
NKARA T URKEY
E-mail address : [email protected] S AIT H ALICIOGLU , D
EPARTMENT OF M ATHEMATICS , A
NKARA U NIVERSITY , 06100 A
NKARA , T
URKEY
E-mail address ::