On A Generalization of Weak Armendariz Rings
aa r X i v : . [ m a t h . R A ] O c t On A Generalization of Weak Armendariz Rings
Mahboubeh Sanaei ∗ , Shervin Sahebi ∗∗ and Hamid H. S. Javadi ∗∗∗ ∗ , ∗∗ Department of Mathematics, Islamic Azad University,Central Tehran Branch, 13185/768, Iran,email: [email protected]; [email protected] ∗∗∗
Department of Mathematics and Computer Science, Shahed University,Tehran, Iran, email: [email protected].
Abstract:
We introduce the notion of J-Armendariz rings, which are a gener-alization of weak Armendariz rings and investigate their properties. We showthat any local ring is J-Armendariz, and then find a local ring that is not weakArmendariz. Moreover, we prove that a ring R is J-Armendariz if and onlyif the n -by- n upper triangular matrix ring T n ( R ) is J-Armendariz. For a ring R and for some e = e ∈ R , we show that if R is an abelian ring, then R is J-Armendariz if and only if eRe is J-Armendariz. Also if the polynomialring R [ x ] is J-Armendariz, then it is proven that the Laurent polynomial ring R [ x, x − ] is J-Armendariz. Mathematics Subject Classification 2010: 16U20, 16S36,16W20Keywords:
Armendariz Ring; Weak Armendariz Ring; J-Armendariz Ring
Throughout this article, R denotes an associative ring with identity. For a ring R , N il ( R ) denotes the set of nilpotents elements in R . In 1997, Rege and Chhawchhariaintroduced the notion of an Armendariz ring. They called a ring R an Armendariz ringif whenever polynomials f ( x ) = a + a x + · · · + a n x n and g ( x ) = b + b x + · · · + b m x m ∈ R [ x ] satisfy f ( x ) g ( x ) = 0 then a i b j = 0 for all i and j . The name ”Armendariz ring”1s chosen because Armendariz [2, Lemma 1] has been shown that reduced ring (thatis a ring without nonzero nilpotent) saisfies this condition. A number of propertiesof the Armendariz rings have been studied in [1, 2, 3, 5, 7, 9]. So far Armendarizrings are generalized in several forms. A generalization of Armendariz rings has beeninvestigated in [4] Liu and Zhao [8] called a ring R weak Armendariz if wheneverpolynomials f ( x ) = a + a x + · · · + a n x n , g ( x ) = b + b x + · · · + b m x m ∈ R [ x ] satisfy f ( x ) g ( x ) = 0, then a i b j ∈ N il ( R ) for all i and j . Recall that the Jacobson radical ofa ring R , is defined to be the intersection of all the maximal left ideals of R . We use J ( R ) to denote the Jacobson radical of R . We call a ring R , J-Armendariz if wheneverpolynomials f ( x ) = a + a x + · · · + a n x n and g ( x ) = b + b x + · · · + b m x m ∈ R [ x ]satisfy f ( x ) g ( x ) = 0 then a i b j ∈ J ( R ) for all i and j . Clearly, weak Armendarizrings are J-Armendariz. Moreover, for an artinian ring, weak Armendariz rings and J-Armendariz rings are the same. But, there exist a J-Armendariz ring that are not weakArmendariz. Thus J-Armendariz rings are a proper generalization of weak Armendarizrings. Furthermore, we prove that the local rings are J-Armendariz. Then we give anexample to show that Local rings are not weak Armendariz in general. In this section J-Armendariz rings are introduced as a generalization of weak Armen-dariz ring.
Definition 2.1.
A ring R is said to be J-Armendariz if for any nonzero polynomial f ( x ) = P ni =0 a i x i and g ( x ) = P mj =0 b j x j ∈ R [ x ] , f ( x ) g ( x ) = 0 , implies that a i b j ∈ J ( R ) for each i, j . Clearly, any Armendariz ring and weak Armendariz ring is J-Armendariz. In thefollowing, we will see that the J-Armendariz rings are not nescessary weak Armendariz.
Example 2.2.
Let A be the 3 by 3 full matrix ring over the power series ring F [[ t ]] over a field F . Let = { M = ( m ij ) ∈ A | m ij ∈ tF [[ t ]] f or ≤ i, j ≤ and m ij = 0 f or i = 3 or j = 3 } C = { M = ( m ij ) ∈ A | m ii ∈ F and m ij = 0 f or i = j } .Let R be the subring of A generated by B and C . Let F = Z . Note that every elementof R is of the form (cid:16) a + f f f a + f
00 0 a (cid:17) for some a ∈ F and f i ∈ tF [[ t ]] ( i = 1 , , , and J ( R ) = tR . Let f ( x ) = P ni =0 (cid:16) a i + f i f i f i a i + f i
00 0 a i (cid:17) x i and g ( x ) = P mj =0 (cid:16) b j + g j g j g j b i + g j
00 0 b j (cid:17) x j ∈ R [ x ] .Assume that f ( x ) g ( x ) = 0 . Then a i b j = 0 for all i and j and so (cid:16) a i + f i f i f i a i + f i
00 0 a i (cid:17)(cid:16) b j + g j g j g j b i + g j
00 0 b j (cid:17) ∈ tR .Hence R is J-Armendariz. Now consider two polynomials over Rf ( x ) = te + te x + te x + te x , g ( x ) = − t ( e + e ) + t ( e + e ) x .Then f ( x ) g ( x ) = 0 , but te t ( e + e ) / ∈ N il ( R ) , and so the ring R is not weakArmendariz. Proposition 2.3.
Let R be a ring and I an ideal of R such that R/I is J-Armendariz.If I ⊆ J ( R ) , then R is J-Armendariz.Proof. Suppose that f ( x ) = a + a x + a x + · · · + a n x n and g ( x ) = b + b x + b x + · · · + b m x m are polynomials in R [ x ] such that f ( x ) g ( x ) = 0. This implies( ¯ a + ¯ a x + ¯ a x + · · · + ¯ a n x n )( ¯ b + ¯ b x + ¯ b x + · · · + ¯ b m x m ) = ¯0,in R/I . Thus ¯ a i ¯ b j ∈ J ( R/I ), And so a i b j ∈ J ( R ). This means that R is a J-Armendarizring. Corollary 2.4.
Let R be any local ring. Then R is J-Armendarz. One may ask if local rings are weak Armendariz, but the following gives a negativeanswer.
Example 2.5.
Let F be a field, R = M ( F ) and R = R [[ t ]] . Consider the ring = { P ∞ i =0 a i t i ∈ R | a ∈ kI f or k ∈ F } ,where I is the identity matrix over F . It is obvious that S is local and so is J-Armendariz. Now for f ( x ) = e t − e tx and g ( x ) = e t + e tx ∈ S [ x ] , we have f ( x ) g ( x ) = 0 , but ( e t ) is not nilpotent in S , and so S is not weak Armendariz. Theorem 2.6.
Let R t be a ring, for each t ∈ I . Then any direct product of rings Q t ∈ I R t , is J-Armendariz if and only if any R t is J-Armendariz.Proof. Suppose that R t is J-Armendariz, for each t ∈ I and R = Q t ∈ I R t . Let f ( x ) g ( x ) = 0 for some polynomials f ( x ) = a + a x + a x + · · · + a n x n , g ( x ) = b + b x + b x + · · · + b m x m ∈ R [ x ], where a i = ( a i , a i , · · · , a i t , · · · ), b j = ( b j , b j , · · · , b j t , · · · )are elements of the product ring R for 1 ≤ i ≤ n and 1 ≤ j ≤ m . Define f t ( x ) = a t + a t x + a t x + · · · + a n t x n , g t ( x ) = b t + b t x + b t x + · · · + b m t x m .From f ( x ) g ( x ) = 0, we have a b = 0 , a b + a b = 0 , · · · a n b m = 0, and this implies a b = a b = · · · = a t b t = · · · = 0 a b + a b = a b + a b = · · · = a t b t + a t b t = · · · = 0 a n b m = a n b m = · · · = a n t b m t = · · · = 0This means that f t ( x ) g t ( x ) = 0 in R t [ x ], for each t ∈ I . Since R t is J-Armendariz foreach t ∈ I , then a i t b j t ∈ J ( R t ). Now the equation Q t ∈ I J ( R t ) = J ( Q t ∈ I R t ), impliesthat a i b j ∈ J ( R ), and so R is J-Armendariz. Conversely, assume that R = Q t ∈ I R t isJ-Armendariz and f t ( x ) g t ( x ) = 0 for some polynomials f t ( x ) = a t + a t x + a t x + · · · + a n t x n , g t ( x ) = b t + b t x + b t x + · · · + b m t x m ∈ R t [ x ], with t ∈ I . Define F ( x ) = a + a x + a x + · · · + a n x n , G ( x ) = b + b x + b x + · · · + b m x m ∈ R [ x ], where a i = (0 , · · · , , a i t , , · · · ), b j = (0 , · · · , , b j t , , · · · ) ∈ R . Since f t ( x ) g t ( x ) = 0, we have F ( x ) G ( x ) = 0. R is J-Armendariz, so a i b j ∈ J ( R ). Therefore a i t b j t ∈ J ( R t ) and so R t is J-Armendariz for each t ∈ I . 4he following example shows that for an Armendariz ring R , every full n -by- n matrixring M n ( R ) over R need not to be J-Armendariz. Example 2.7.
Let F be a field and R = M ( F ) . If f ( x ) = (cid:0) (cid:1) − (cid:0) (cid:1) x and g ( x ) = (cid:0) (cid:1) + (cid:0) − − (cid:1) x , then f ( x ) g ( x ) = 0 . But (cid:0) (cid:1)(cid:0) (cid:1) = (cid:0) (cid:1) is not in J ( R ) .Thus R is not J-Armendariz. Let R and S be two rings and M be an ( R, S )-bimodule. This means that M is aleft R -module and a right S -module such that ( rm ) s = r ( ms ) for all r ∈ R , m ∈ M ,and s ∈ S . Given such a bimodule M we can form T = (cid:0) R M S (cid:1) = (cid:8)(cid:0) r m s (cid:1) : r ∈ R, m ∈ M, s ∈ S (cid:9) and define a multiplication on T by using formal matrix multiplication: (cid:0) r m s (cid:1)(cid:0) r ′ m ′ s ′ (cid:1) = (cid:0) rr ′ rm ′ + ms ′ ss ′ (cid:1) . This ring construction is called triangular ring T . Proposition 2.8.
Let R and S be two rings and T be the triangular ring T = (cid:0) R M S (cid:1) (where M is an ( R, S ) -bimodule). Then the rings R and S are J-Armendariz if andonly if T is J-Armendariz.Proof. Let R and S be J-Armendarz, and f ( x ) = (cid:0) r m s (cid:1) + (cid:0) r m s (cid:1) x + · · · + (cid:0) r n m n s n (cid:1) x n , g ( x ) = (cid:0) r ′ m ′ s ′ (cid:1) + (cid:0) r ′ m ′ s ′ (cid:1) x + · · · + (cid:0) r ′ m m ′ m s ′ m (cid:1) x m ∈ T [ x ]satisfy f ( x ) g ( x ) = 0. Define f r ( x ) = r + r x + · · · + r n x n , g r ( x ) = r ′ + r ′ x + · · · + r ′ m x m ∈ R [ x ]and f s ( x ) = s + s x + · · · + s n x n , g s ( x ) = s ′ + s ′ x + · · · + s ′ m x m ∈ S [ x ] . f ( x ) g ( x ) = 0, we have f r ( x ) g r ( x ) = f s ( x ) g s ( x ) = 0, and since R and S areJ-Armendariz then r i r ′ j ∈ J ( R ) and s i s ′ j ∈ J ( S ) for each 1 ≤ i ≤ n , 1 ≤ j ≤ m .Now from the fact J ( T ) = (cid:0) J ( R ) M J ( S ) (cid:1) , we obtain that (cid:0) r i m i s i (cid:1)(cid:0) r ′ j m ′ j s ′ j (cid:1) ∈ J ( T ) forany i, j . Hence T is a J-Armendariz ring. Conversely, let T be a J-Armendariz ring, f r ( x ) = r + r x + · · · + r n x n , g r ( x ) = r ′ + r ′ x + · · · + r ′ m x m ∈ R [ x ], such that f r ( x ) g r ( x ) = 0, and f s ( x ) = s + s x + · · · + s n x n , g s ( x ) = s ′ + s ′ x + · · · + s ′ m x m ∈ S [ x ],such that f s ( x ) g s ( x ) = 0. If f ( x ) = (cid:0) r s (cid:1) + (cid:0) r s (cid:1) x + · · · + (cid:0) r n s n (cid:1) x n and g ( x ) = (cid:0) r ′ s ′ (cid:1) + (cid:0) r ′ s ′ (cid:1) x + · · · + (cid:0) r ′ m s ′ m (cid:1) x m ∈ T [ x ]Then from f r ( x ) g r ( x ) = 0 and f s ( x ) g s ( x ) = 0 it follows that f ( x ) g ( x ) = 0. Since T is a J-Armendariz ring, (cid:0) r i s i (cid:1)(cid:0) r ′ j s ′ j (cid:1) ∈ J ( T ) = (cid:0) J ( R ) 00 J ( S ) (cid:1) . Thus r i r ′ j ∈ J ( R ) and s i s ′ j ∈ J ( S ) for any i, j . This shows that R and S are J-Armendariz.Given a ring R and a bimodule R M R , the trivial extension of R by M is the ring T ( R, M ) = R L M with the usual addition and the multiplication( r , m )( r , m ) = ( r r , r m + m r ) . This is isomorphic to the ring of all matrices (cid:0) r m r (cid:1) , where r ∈ R and m ∈ M and theusual matrix operations are used. Corollary 2.9.
A ring R is J-Armendariz if and only if the trivial extension T ( R, R ) is a J-Armendariz ring. Corollary 2.10.
A ring R is J-Armendariz if and only if, for any n , T n ( R ) is J-Armendariz. Corollary 2.11. If R is a Armendariz ring then, for any n , T n ( R ) is a J-Armendarizring. Recall that a ring R is said to be abelian if every idempotent of it is central. Armen-dariz rings are abelian [7, Lemma 7], but the next example shows that weak Armendariz6nd J-Armendariz rings need not to be abelian in general. Example 2.12.
Let F be a field. By Corollary 2.11, R = T ( F ) is a J-Armendarizring. We see that (cid:0) (cid:1) is an idempotent element in R , that is not central. So R is notan abelian ring. Proposition 2.13.
Let R be a J-Armendariz ring. Then for any idempotent e of R , eRe is J-Armendariz. The converse holds if R is an abelian ring.Proof. Let f ( x ) = P ni =0 a i x i , g ( x ) = P mj =0 b j x j ∈ ( eRe )[ x ] be such that f ( x ) g ( x ) = 0.Since R is J-Armendariz and a i , b j ∈ eRe ⊆ R , then we have a i b j ∈ J ( R ) ∩ eRe = J ( eRe ). This means that eRe is J-Armendariz. Conversely, let eRe be a J-Armendarizring and f ( x ) = P ni =0 a i x i , g ( x ) = P mj =0 b j x j ∈ R [ x ], such that f ( x ) g ( x ) = 0. Bythe hypothesis, 0 = ef ( x ) eg ( x ) e ∈ ( eRe )[ x ], and since eRe is J-Armendariz, we have a i b j ∈ J ( eRe ) = J ( R ) ∩ eRe . Thus R is J-Armendariz.In [1] it is proven that a ring R is Armendariz if and only if its polynomial ring R [ x ]is Armendariz. More generally, we can get the following result. Theorem 2.14.
If the ring R [ x ] is J-Armendariz, then R is J-Armendariz. The con-verse holds if J ( R )[ x ] ⊆ J ( R [ x ]) .Proof. Suppose that R [ x ] is a J-Armendariz ring. Let f ( y ) = P ni =0 a i y i and g ( y ) = P mj =0 b j y j be nonzero plynomials ∈ R [ y ], such that f ( y ) g ( y ) = 0. Since R [ x ] is J-Armendariz and R ⊆ R [ x ], we have a i b j ∈ R ∩ J ( R [ x ]), and so R is J-Armendariz.Conversely, suppose that R is J-Armendariz and J ( R )[ x ] ⊆ J ( R [ x ]). Let F ( y ) = f + f y + · · · + f n y n and G ( y ) = g + g y + · · · + g m y m be polynomials in R [ x ][ y ],with F ( y ) G ( y ) = 0. We also let f i ( x ) = a i + a i x + a i x + · · · + a i ωi x ω i and g j ( x ) = b j + b j x + b j x + · · · + b j νj x ν i ∈ R [ x ] for each 0 ≤ i ≤ n and 0 ≤ j ≤ m . Take apositive integer t suhc that t ≥ deg ( f ( x ))+ deg ( f ( x ))+ · · · + deg ( f n ( x ))+ deg ( g ( x ))+ deg ( g ( x )) + · · · + deg ( g m ( x )), where the degree is as polynomials in x and the degreeof zero polynomial is taken to be 0. Then F ( x t ) = f + f x t + · · · + f n x tn and G ( x t ) =7 + g x t + · · · + g m x tm ∈ R [ x ] and the set of coefficients of the f i ’s (resp. g j ’s)equals the set of coefficients of the F ( x t ) (resp. G ( x t )). Since F ( y ) G ( y ) = 0, then F ( x t ) G ( x t ) = 0. So a is i b jr j ∈ J ( R ), where 0 ≤ s i ≤ ω i , 0 ≤ r j ≤ ν j . By hypothesis wehave J ( R )[ x ] ⊆ J ( R [ x ]), and so f i g j ∈ J ( R [ x ]). It implies that R is J-Armendariz. Proposition 2.15.
Let R is a J-Armendariz ring and S denotes a multiplicativelyclosed subset of a ring R consisting of central regular elements. Let S − R denotes thelocalization of R at S . Then S − R is a J-Armendariz ring.Proof. Suppose that R is a J-Armendariz ring. Let F ( x ) = P ni =0 ( α i ) x i and G ( x ) = P mj =0 ( β j ) x j be nonzero polynomials in ( S − R )[ x ] such that F ( x ) G ( x ) = 0, where α i = a i u − , β j = b j v − , with a i , b j ∈ R and u, v ∈ S . Since S is contained in the center of R ,we have F ( x ) G ( x ) = ( uv ) − ( a + a x + a x + · · · + a n x n )( b + b x + b x + · · · + b m x m ) = 0.Let f ( x ) = a + a x + a x + · · · + a n x n and g ( x ) = b + b x + b x + · · · + b m x m .Then f ( x ) and g ( x ) are nonzero polynomials in R [ x ] with f ( x ) g ( x ) = 0. Since R isJ-Armendariz, then a i b j ∈ J ( R ). It means that α i β j ∈ J ( S − R ), concluding that S − R is J-Armendariz. Corollary 2.16.
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