Skew Monoid Rings over Zip Rings

Transcription

1 International Journal of Algebra, Vol. 4, 2010, no. 21, Skew Monoid Rings over Zip Rings Amit Bhooshan Singh, M. R. Khan and V. N. Dixit Department of Mathematics Jamia Millia Islamia (Central University) New Delhi , India vn Abstract Let R be a ring and G is a u.p. monoid (unqiue product monoid). Assume that there is a monoid homomorphism σ : G Aut(R). Suppose that σ is weakly rigid then the skew monoid ring R G is right zip ring if and only if R is right zip ring. Mathematics Subject Classification: Primary: 16D25, 16P60; Secondary: 16S35, 16S36 Keywords: Skew monoid ring, weakly rigid ring, zip ring 1 Introduction Throughout this paper R denotes associated ring with identity and all modules are unitary. According to Krempa [13] an endomorphism of a ring R is called rigid if aα(a) =0 a = 0, for a R. We say a ring R is α-rigid if there exists a rigid endomorphism σ of R. Recall that a ring R is reduced if a 2 =0 a = 0, for a R. Note that any rigid endomorphism of a ring R is monomorphism and α-rigid rings are reduced rings. In [8], Faith introduced that a ring R is right zip ring if the right annihilator r R (V ) of a subset V of R is zero then there exists a finite subset V 0 V s.t. r R (V ) is zero. Equivalently for a left ideal L of R with r R (L) = 0 there exists a finitely generated left ideal L L s.t. r R (L) =0. R is a zip ring if it is right and left zip ring. The concept of zip ring was introduced by Zelmenwitz in [19] and appeared in various papers [2-5]. Zelmanwitz stated that any ring satisfying the descending chain condition on annihilators is a right zip ring (although not so called at that time), but the converse does not hold. In [3], Beachy and Blair have shown that if R is

2 1032 A. B. Singh, M. R. Khan and V. N. Dixit commutative zip ring then polynomial ring R[x] over zip ring R is right zip ring. In [8], faith proved that if a ring R is commutative ring and G is finite abelian group then the group ring R[G] ofg over R is zip ring. In [5], Cedo has shown that if R is a commutative zip ring then the n n full matrix ring mat n (R) over R is zip ring. In [10], C.Y. Hong etc. proved that a ring R is right zip ring if and only if autm n (R) is right zip ring. Let R be an Armendariz ring then R is right zip ring iff R[x] is right zip ring R is commutative ring and G is a u.p. monoid that contains an infinite cyclic submonoid then R is a zip ring iff R[G] is zip ring. In [9], W. Cartes has shown that let σ be an automorphism of R and R satisfies SA1 condition then R is right zip ring iff R[x, σ] is right zip ring iff R[x, x 1,σ] is right zip ring. In this paper we investigated that a ring R is right zip ring iff R G is right zip ring, when M is u.p. monoid and σ is weakly rigid. 2 Main Results Definition 2.1. Let σ be an automorphism of a ring R. We define σ to be weakly rigid if ab =0implies aσ(b) =0or σ(a)b =0for any a, b R. A monoid homomorphism σ from a monoid G into the group of automorphism of R, x α x is called weakly rigid if α x Aut(R) is weakly rigid for every x G. Example 2.2 ([Example 2.4, 15]). (i) If for every x G, α x = id, then α is a weakly rigid. (ii) Let σ be an endomorphism of R according to [11] and [13], σ is called a rigid endomorphism if aσ(a) =0implies a =0for a R. A ring R is said to be σ-rigid if there exists a rigid endomorphism σ of R. Clearly every rigid endomorphism is monomorphism and every σ-rigid ring is reduced. Let σ be rigid automorphism of it was shown in [12] that if ab =0then aσ n (b) =0or σ n (a)b =0for any positive integer n. Thus the map α : Z aut(r) :σ(x) =σ ( x is weakly ) rigid. Let β be a rigid automorphism z z of a ring R 0 and S =. Set R 0 z 1 = R 0 S, the direct sum of rings R 0 and S. Define an endomorphism σ of R via σ(r, s) =(β(r),s) where r R and s S. Then it is easy to see that σ is weakly rigid and so the map σ : Z Aut(R 1 ):σ(x) =σ x is weakly rigid but σ is not rigid. {( ) a p (iii) Let R = a Z, p Q} where Q is a set of al rational numbers. 0 a Let σ : R R be an automorphism defined by (( )) ( ) a 0 a p/2 σ = 0 a 0 a

3 Skew monoid rings over zip rings 1033 then σ is not rigid but σ is weakly rigid and the map α : Z Aut(R) : σ(x) =σ x is weakly rigid. {( ) a b (iv) Let R is reduced ring. Consider the ring T = a, b R}.Let (( 0 )) a ( ) a b a b σ : T T be an automorphism defined by T = so 0 a 0 a σ is not rigid but σ is weakly rigid and so the map σ : Z Aut(R) : α(x) =σ x is weakly rigid. Recall that a monoid G is called unique product monoid (u.p. monoid) if for any two nonempty finite subsets A, B G, there exists an element g G uniquely presented in the form ab where a A and b B. The class of u.p. monoids is quite large and important (See [4], [15], [16]). For examples this class includes the right or left ordered monoids, submonoids, of a free group and torsion free nilpotent groups. Every u.p. monoid G has no non-unity element of finite order. Let R be a ring and G be a u.p. monoid. Assume that G acts on R by means of a group of R we denote the image of r R under g G by σ g (r). We can form a skew monoid ring R G (induced by the monoid homomorphism σ). The skew monoid ring R G is a ring which as a left R-module is free with basis G and multiplication defined by the rule gr = σ g (r)g. Theorem 2.3 ([12, Theorem 3]). Let R be a ring, G be a u.p. monoid and A a right ideal of R G. Ifr R G (A) 0. Then r R (A) 0. Theorem 2.4. Let R be a ring and G is v.p. monoid. Let σ is weakly rigid, then skew monoid ring R G induced by σ is right zip ring iff R is right zip ring. Proof. Suppose R is right zip ring and U R G with r R G (U) = 0. Let V be the set of coefficients of elements of U so V R. If any element p r R (V ), then Vp= 0 implies vp =0 v V. Now take any element β = v 0 g 0 + v 1 g 1 + v 2 g v n g n U where v i V and g i G. so βp = (v 0 g 0 + v 1 g v n g n )p = v 0 g 0 p + v 1 g 1 p v n gn p = v 0 σ g0 (p)g 0 + v 1 σ g1 (p)g v n σ gn (p)g n. Since σ is weakly rigid so if vp = 0 implies vσ g (p) =0 v V and g G. Thus βp = i.e. βp = 0 implies p r R G (U) =0sop = 0 thus p r R (v) = 0. Since R is rigid zip and V R such that r R (v 0 )=0. For any p V 0, there exists α p U such that some coefficients of α p is p. Let U 0 is minimal subset of U such that α p U 0 for each. Then U 0 is finite subset

4 1034 A. B. Singh, M. R. Khan and V. N. Dixit of U. Let V 1 be a set of all coefficients of elements in U 0. Then V 0 V 1 and so r R (V 1 ) r R (V 0 ) = 0. Thus r R (V 1 ) = 0. Now only to show that r R G (U 0 )=0. Suppose r R G (U 0 ) 0 so there exists 0 α = a 0 h 0 + a 1 h a n h n r R G (U 0 ) then U 0 α = 0 so by the Theorem 2.3 U 0 a 0 = 0, where a 0 0, then a 0 r R (U 0 ). Since σ is weakly rigid so a 0 r R (V 1 ). But r R (V 1 ) = 0 therefore a 0 = 0, which is contraction so r R G (U 0 ) = 0. Hence R G is skew monoid ring. Conversely, assume that R G is right zip ring and let any subset V R with r R (V ) = 0. Let any element α = a 0 h 0 + a 1 h a n h n r R G (V ) where a i R and h i G. Then Vα= 0 i.e vα =0 v V.Thusva i = 0 for all j =0, 1, 2,...,n implies a i r R (V )=0soa i = 0 for all i =0, 1, 2,...,n. Thus α = 0 that means r R G (V ) = 0. Since R G is right zip rings so there exists a finite subset. V 0 V such that r R G (V 0 ) = 0. Therefore r R (V 0 )= r R G (V 0 ) R = 0 implies r R (v 0 ) = 0. Hence R is right zip ring. In [18], Rage and Chhawchharia introduced that a ring R is called Aremendariz if a i b j = 0 for all i and j, whenever polynomials f(x) = m a i x i, n and g(x) = b j x i R[x] satisfy f(x)g(x) = 0. suppose that σ is an endomorphism of R. A ring R satisfies condition SA1 if for f(x) = m a i x i j=0 and g(x) = n j=0 b j x j R[x, j]f(x)g(x) = 0 implies a i σ i (b j ) = 0 for all 0 i m and 0 j n. Corollary 2.5 ([10, Theorem 16]). Suppose that R is a commutative ring and G is a u.p. monoid that contains infinite cyclic submonoids then R is zip ring if and only if R[G] is zip ring. Corollary 2.6 ([8]). A commutative ring R is zig ring if and only if R[x] is zip ring. Corollary 2.7 ([10, Theorem 11]). Let R is Armendariz ring then R is right zip ring if and only if R[x] is right zip ring. Corollary 2.8 ([9, Theorem 2.8]). Let σ be an automorphism of R and suppose R satisfies SA1. Then the following are equivalent. (1) R is right zip ring. (2) R[x, σ] is right zip ring. (3) R[x, x 1,σ] is right zip ring. Corollary 2.9. Let R be a right Mccoy ring. Then R[x] is right zip ring if and only if R is right zip ring. i=0 i=0

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