EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 7, No. 4, 2014, 387-394 ISSN 1307-5543 – www.ejpam.com Skew-Laurent rings over σ(∗)-rings V. K. Bhat School of Mathematics, SMVD University, P/o SMVD University, Katra, J and K, India- 182320 Abstract. Let R be an associative ring with identity 1 6= 0, and σ an endomorphism of R. We recall σ(∗) property on R (i.e. aσ(a) ∈ P(R) implies a ∈ P(R) for a ∈ R, where P(R) is the prime radical of R). Also recall that a ring R is said to be 2-primal if and only if P(R) and the set of nilpotent elements of R coincide, if and only if the prime radical is a completely semiprime ideal. It can be seen that a σ(∗)-ring is a 2-primal ring. Let R be a ring andσ an automorphism of R. Then we know thatσ can be extended to an automorphism (say σ) of the skew-Laurent ring R[x , x−1;σ]. In this paper we show that if R is a Noetherian ring and σ is an automorphism of R such that R is a σ(∗)-ring, then R[x , x−1;σ] is a σ(∗)-ring. We also prove a similar result for the general Ore extension R[x;σ,δ], where σ is an automorphism of R and δ a σ-derivation of R. 2010 Mathematics Subject Classifications: 16-XX; 16N40, 16P40, 16S36. Key Words and Phrases: Minimal prime, prime radical, automorphism, σ(∗)-ring 1. Introduction A ring R always means an associative ring with identity 1 6= 0. The set of prime ideals of R is denoted by Spec(R). The sets of minimal prime ideals of R is denoted by Min.Spec(R). Prime radical and the set of nilpotent elements of R are denoted by P(R) and N(R) respectively. Let R be a ring and σ an automorphism of R. Let I be an ideal of R such that σm(I) = I for some m ∈ N (where N is the set of positive integers). We denote ∩m i=1σ i(I) by I0. The field of rational numbers is denoted by Q and the field of real numbers is denoted by R unless otherwise stated. This article concerns the study of skew-Laurent rings over σ(∗)-rings, where σ is an auto- morphism of R. Email address: vijaykumarbhat2000@yahoo.com http://www.ejpam.com 387 c© 2014 EJPAM All rights reserved. V. Bhat / Eur. J. Pure Appl. Math, 7 (2014), 387-394 388 σ(∗)-rings Recall that in Krempa [8], a ring R is called σ-rigid if there exists an endomorphism σ of R with the property that aσ(a) = 0 implies a = 0 for a ∈ R. In [9], Kwak defines a σ(∗)-ring R to be a ring in which aσ(a) ∈ P(R) implies a ∈ P(R) for a ∈ R. Example 1. Let R = � F F 0 F � , where F is a field. Then P(R) = � 0 F 0 0 � . Let σ : R→ R be defined by σ � � a b 0 c � � = � a 0 0 c � . Then it can be seen that σ is an endomorphism of R and R is a σ(∗)-ring. 2-primal Rings We do not want to talk about 2-primal rings, but because of a close relation between a σ(∗)-ring and a 2-primal ring, we have the following: Recall that a ring R is 2-primal if and only if N(R) = P(R), i.e. if the prime radical is a completely semiprime ideal. An ideal I of a ring R is called completely semiprime if a2 ∈ I implies a ∈ I for a ∈ R. We note that a commutative ring is 2-primal and so is a reduced ring. 2-primal rings have been studied in recent years and the 2-primal property is being studied for various types of rings. In [10], Greg Marks discusses the 2-primal property of R[x;σ,δ], where R is a local ring, σ is an automorphism of R and δ is a σ-derivation of R. He has proved that when R is a local ring with a nilpotent maximal ideal, the Ore extension R[x;σ,δ] will or will not be 2-primal depending on the δ-stability of the maximal ideal of R. In [9], Kwak establishes a relation between a 2-primal ring and a σ(∗)-ring. It has been proved that if R is a ring and σ an endomorphism of R such that σ(P(R)) ⊆ P(R), then R is a σ(∗)-ring implies that R is 2-primal. Therefore, we see that if R is a Noetherian ring and σ an automorphism of R, then R is a σ(∗)-ring implies that R is 2-primal. The following example shows that if R is a Noetherian ring, then even R[x] need not be 2-primal. Example 2. Let R = M2(Q), the set of 2 × 2 matrices over Q. Then R[x] is a prime ring with non-zero nilpotent elements and, so can not be 2-primal. Skew Polynomial Rings Let R be a ring, σ be an endomorphism of R and δ a σ-derivation of R. Recall that δ is an additive map δ : R→ R such that δ(ab) = δ(a)σ(b) + aδ(b), for all a, b ∈ R. Example 3. Let σ be an automorphism of a ring R and δ : R→ R any map. Let φ : R→ M2(R) defined by φ(r) = � σ(r) 0 δ(r) r � , for all r ∈ R be a homomorphism. Then δ is a σ-derivation of R. V. Bhat / Eur. J. Pure Appl. Math, 7 (2014), 387-394 389 Recall that the skew polynomial ring (Ore extension) R[x;σ,δ] is the usual ring of polyno- mials with coefficients in R, in which multiplication is subject to the relation ax = xσ(a)+δ(a) for all a ∈ R. We take any f (x) ∈ R[x;σ,δ] to be of the form f (x) = ∑n i=0 x iai . We denote R[x;σ,δ] by O(R). If I is an ideal of R such that σ(I) = I and δ(I) ⊆ I , then O(I) denotes I[x;σ,δ], which is an ideal of O(R). Skew-Laurent Rings Recall that R[x , x−1;σ] is the usual ring of Laurent polynomials with coefficients in R, in which multiplication is subject to the relation ax = xσ(a) for all a ∈ R. We take any f (x) ∈ R[x , x−1;σ] to be of the form f (x) = ∑n i=−m x iai . We denote R[x , x−1;σ] by L(R). If an ideal I of a ring R is σ-stable (i.e. σ(I) = I), then we denote as usual I[x , x−1;σ] by L(I). We also note that ifσ is an automorphism of R, then it can be extended to an automorphism (say σ) of R[x , x−1;σ] such that σ(x) = x; i.e. σ(Σn i=−m x iai) = Σn i=−m x iσ(ai). The study of skew polynomial rings and skew-Laurent rings has been of interest to many authors. For example [1, 6, 7, 9]. In this paper we prove the following results: Theorem 2: Let R be a Noetherian ring and σ an automorphism of R. Then R is a σ(∗)-ring if and only if R[x , x−1;σ] is a σ(∗)-ring. Theorem 3: Let R be a Noetherian ring which is also an algebra over Q. Let σ be an automor- phism of R such that R is a σ(∗)-ring and δ a σ-derivation of R such that σ(δ(a)) = δ(σ(a)) for all a ∈ R. Then R[x;σ,δ] is a σ(∗)-ring. 2. Preliminaries We begin this section with the following Proposition: Proposition 1. Let R be a ring and σ an automorphism of R. Then R is a σ(∗)-ring implies R is 2-primal. Proof. Let a ∈ R be such that a2 ∈ P(R). Then aσ(a)σ(aσ(a)) = aσ(a)σ(a)σ2(a) ∈ σ(P(R)) = P(R). Therefore aσ(a) ∈ P(R) and hence a ∈ P(R). The following example shows that there exists an endomorphism σ of a ring R such that the converse of the above Proposition does not hold. Example 4. Let R= F[x], F a field. Then R is a commutative domain, and therefore is 2-primal with P(R) = 0. Let σ : R→ R be defined by σ( f (x)) = f (0). Let f (x) = xa, 0 6= a ∈ F. Then f (x)σ( f (x)) ∈ P(R), but f (x) /∈ P(R). Therefore R is not a σ(∗)-ring. V. Bhat / Eur. J. Pure Appl. Math, 7 (2014), 387-394 390 Before we give a characterization of a Noetherian σ(∗)-ring, we require the following: Recall that an ideal P of a ring R is completely prime if R/P is a domain, i.e. ab ∈ P implies a ∈ P or b ∈ P for a, b ∈ R (McCoy [11]). Note that a completely prime ideal is a prime ideal, but the converse need not be true. For example, let R= � Z Z Z Z � = M2(Z). If p is a prime number, then the ideal P = M2(pZ) is a prime ideal of R, but is not strongly prime, since for a = � 1 0 0 0 � and b = � 0 0 0 1 � we have ab ∈ P, even though a /∈ P and b /∈ P. Proposition 2 (Proposition 2.1 of Bhat [6]). Let R be a Noetherian ring, andσ an automorphism of R. Then R is a σ(∗)-ring if and only if for each minimal prime U of R, σ(U) = U and U is a completely prime ideal of R. Proof. To make the article self contained, we give a proof (a modified one): Let R be a Noetherian ring such that for each minimal prime U of R, σ(U) = U and U is completely prime ideal of R. Let a ∈ R be such that aσ(a) ∈ P(R) = ∩n i=1Ui , where Ui are the minimal primes of R. Now for each i, a ∈ Ui or σ(a) ∈ Ui as Ui are completely prime. Now σ(a) ∈ Ui = σ(Ui) implies that a ∈ Ui . Therefore a ∈ P(R). Hence R is a σ(∗)-ring. Conversely, suppose that R is aσ(∗)-ring and let U = U1 be a minimal prime ideal of R. Now by Proposition 1, P(R) is completely semiprime. Now Min.Spec(R) is finite by Theorem (2.4) of Goodearl and Warfield [7]. Let U2, U3, . . . , Un be the other minimal primes of R. Suppose that σ(U) 6= U . Then σ(U) is also a minimal prime ideal of R. Renumber so that σ(U) = Un. Let a ∈ ∩n−1 i=1 Ui . Then σ(a) ∈ Un, and so aσ(a) ∈ ∩n i=1Ui = P(R). Therefore a ∈ P(R), and thus ∩n−1 i=1 Ui ⊆ Un, which implies that Ui ⊆ Un for some i 6= n, which is impossible. Hence σ(U) = U . Now since a σ(∗)-ring is 2-primal, minimal prime ideals are completely prime. Hence U is completely prime. Note that in above Theorem the condition of completely primeness of minimal prime ideals can not be deleted. Towards this we have the following: Remark 1. Let R be a Noetherian ring and σ an automorphism of R such that σ(U) = U for each minimal prime ideal U of R. Then R need not be a σ(∗)-ring (Example 4). 3. Skew-Laurent Rings Over σ(∗)-rings Goodearl and Warfield proved in (2ZA) of [7] that if R is a commutative Noetherian ring, and if σ is an automorphism of R, then an ideal I of R is of the form P ∩ R for some prime ideal P of R[x , x−1;σ] if and only if there is a prime ideal S of R and a positive integer m with σm(S) = S, such that I = ∩σi(S), i = 1, 2, . . . , m. We note that if R is a Noetherian ring, then as mentioned above, Min.Spec(R) is finite. Now if σ is an automorphism of R, then σ j(U) ∈ Min.Spec(R) for any U ∈ Min.Spec(R) for V. Bhat / Eur. J. Pure Appl. Math, 7 (2014), 387-394 391 all j ∈ N. Therefore, there exists some m ∈ N such that σm(U) = U for all U ∈ Min.Spec(R). We denote ∩m i=1σ i(U) by U0. We now have the following: Theorem 1. Let R be a Noetherian ring andσ an automorphism of R. Then P ∈ Min.Spec(L(R)) if and only if there exists U ∈ Min.Spec(R) such that L(P ∩ R) = (P ∩ R)[x , x−1;σ] = P and P ∩ R= U0. Proof. See Theorem (2.4) of Bhat [1]. As mentioned in the introduction, we note that if σ is an automorphism of R, then it can be extended to an automorphism (say σ) of R[x , x−1;σ] such that σ(x) = x; i.e. σ(Σn i=−m x iai) = Σn i=−m x iσ(ai). With this we are now in a position to prove the following Theorem: Theorem 2. Let R be a Noetherian ring and σ an automorphism of R. Then R is a σ(∗)-ring if and only if L(R) = R[x , x−1;σ] is a Noetherian σ(∗)-ring. Proof. Let R be a Noetherian ring, σ an automorphism of R such that R is a σ(∗)-ring and δ a σ-derivation of R. We shall prove that O(R) = R[x;σ,δ] is a Noetherian σ(∗)-ring. For this we will show that any minimal P ∈ Min.Spec(O(R)) is completely prime and σ(P) = P. Let P ∈ Min.Spec(O(R)). Then by Theorem 1, there exists U ∈ Min.Spec(R) such that P = U0[x , x−1;σ]. Now R is aσ(∗)-ring implies thatσ(U) = U by Proposition 2, and therefore U0 = U . So P = U[x , x−1;σ] and thus σ(P) = P. We now show that P = U[x , x−1;σ] is completely prime. Now σ can be extended to an automorphism of R/U in a natural way. We note that O(R)/P ∼= (R/U)[x , x−1;σ], and since U is completely prime, R/U is a domain and so (R/U)[x , x−1;σ] is also a domain. Hence P = U[x , x−1;σ] is completely prime. Thus σ(P) = P and P is completely prime for all P ∈ Min.Spec(L(R)). Moreover L(R) = R[x , x−1;σ] is Noetherian by Theorem (1.17) of Goodearl and Warfield [7]. Hence by Proposition 2 R[x , x−1;σ] is a σ(∗)-ring. Conversely let L(R) = R[x , x−1;σ] be a σ(∗)-ring. Let U ∈ Min.Spec(R). Then Theorem 1 implies that L(U0) ∈ Min.Spec(L(R)). Now L(R) be a σ(∗)-ring implies that σ(L(U0)) = L(U0) and L(U0) is completely prime ideal of L(R). Now there is an embedding R/(L(U0) ∩ R)→ L(R)/L(U0). Since L(R)/L(U0) is an integral domain, so is R/(L(U0) ∩ R). Therefore, U0 = L(U0) ∩ R) is a completely prime ideal of R. Now U0 ⊆ U implies that U0 = U . So σ(U) = U and U is a completely prime ideal of R. Hence by Proposition 2 R is a σ(∗)-ring. Remark 2. i) Let R be a Noetherian ring and σ an automorphism of R such that R is a σ(∗)-ring. Then R[x , x−1;σ] is a σ(∗)-ring. Therefore, Proposition 1 implies that R[x , x−1;σ] is 2-primal. V. Bhat / Eur. J. Pure Appl. Math, 7 (2014), 387-394 392 ii) If R is 2-primal Noetherian ring, then R[x , x−1;σ] need not be 2-primal. For example consider Z2 and let R= Z2⊕Z2. Then R is a commutative reduced ring with P(R) = 0, and therefore R is 2-primal. Define σ : R→ R by σ(a, b) = (b, a). Then it can be seen that P(R[x , x−1;σ]) = 0, but P(R[x , x−1;σ]) is not completely semiprime as ((1, 0)x)2 = 0= P(R[x , x−1;σ]), but (1,0)x /∈ P(R[x , x−1;σ]). Thus R[x , x−1;σ] is not 2-primal. 4. Skew Polynomial Rings Over σ(∗)-rings Let σ be an endomorphism of a ring R and δ a σ-derivation of R such that σ(δ(a)) = δ(σ(a)) for all a ∈ R. Then σ can be extended to an endomorphism (say σ) of R[x;σ,δ] by σ( ∑m i=0 x iai) = ∑m i=0 x iσ(ai). Also δ can be extended to a σ-derivation (say δ) of R[x;σ,δ] by δ( ∑m i=0 x iai) = ∑m i=0 x iδ(ai). Example 5 (Example 2.13 of Bhat [5]). Let R= R×R, σ : R→ R defined by σ((a, b)) = (b, a) for a, b ∈ R. Then σ is an automorphism of R. Let now r ∈ R. Define δr : R→ R by δr((a, b)) = (a, b)r − rσ((a, b)) for a, b ∈ R. Then δ is a σ-derivation. Now for any (u, v) ∈ R, σ(δr((u, v))) =σ((u, v)r − rσ((u, v))) =σ((u, v)r − r(v, u)) =σ((ur, vr)−σ(vr, ur)) =(vr, ur)− (ur, vr)). Also δr(σ((u, v))) =δr(v, u) =(v, u)r − rσ((v, u)) =(v, u)r − r(u, v) =(vr, ur)− (ur, vr)). Therefore σ(δ((u, v))) = δ(σ((u, v))) for all (u, v) ∈ R. Remark 3. We note that if σ(δ(a)) 6= δ(σ(a)) for all a ∈ R, then the above does not hold. For example let f (x) = x l and g(x) = x p, a, b ∈ R. Then δ( f (x)g(x)) = x2{δ(σ(l))σ(p) +σ(l)δ(p)}+ x{δ2(l)σ(p) +δ(l)σ(p)}, but δ( f (x))σ(g(x)) + f (x)δ(g(x)) = x2{σ(δ(l))σ(p) +σ(l)δ(p)}+ x{δ2(l)σ(p) +δ(l)σ(p)}. So, δ( f (x)g(x)) 6= δ( f (x))σ(g(x)) + f (x)δ(g(x)), i.e. δ is not a δ-derivation. REFERENCES 393 With this we now prove the following: Theorem 3. Let R be a Noetherian ring which is also an algebra over Q. Let σ be an automor- phism of R and δ a σ-derivation of R such that σ(δ(a)) = δ(σ(a)) for all a ∈ R. Further let P ∈ Min.Spec(O(R)) implies that P ∩ R ∈ Min.Spec(R). Then R is a σ(∗)-ring implies that O(R) = R[x;σ,δ] is a Noetherian σ(∗)-ring. Proof. Let R be a Noetherian ring and σ an automorphism of R such that R is a σ(∗)-ring. We shall prove that O(R) = R[x;σ,δ] is a Noetherian σ(∗)-ring. For this we will show that any minimal P ∈ Min.Spec(O(R)) is completely prime and σ(P) = P. Let P ∈ Min.Spec(O(R)). Now P ∩ R ∈ Min.Spec(R) and R is a σ(∗)-ring implies that σ(P ∩ R) = P ∩ R and P ∩ R is a completely prime ideal of R. Now Proposition (2.1) of Bhat [2] implies that δ(P ∩ R) ⊆ P ∩ R. Now Theorem (2.4) of Bhat [4] implies that O(P ∩ R) is a completely prime ideal of O(R). Now O(P ∩R) ⊆ P implies that O(P ∩R) = P as P is minimal. Now σ(P ∩ R) = P ∩ R implies that σ(P) = P. Thus σ(P) = P and P is completely prime for all P ∈ Min.Spec(O(R)). Moreover O(R) = R[x;σ,δ] is Noetherian by Theorem (1.12) of Goodearl and Warfield [7]. Hence by Proposition 2 R[x;σ,δ] is a σ(∗)-ring. We note that the condition that P ∈ Min.Spec(O(R)) implies that P ∩ R ∈ Min.Spec(R) can not be ignored as follows: Let R = Q×Q. Let σ : R→ R be defined by σ((a, b)) = (b, a) and δ = 0. Then P = 0 is a prime ideal of O(R), but P ∩ R is not a prime ideal of R. We have not been able to prove the converse part of the above result. The main reason being that a generalization of Theorem 1 in terms of O(R) is not known. The known towards this is: Let R be a Noetherian ring which is also an algebra overQ. Let σ be an automorphism of R and δ a σ-derivation of R. Then U ∈ Min.Spec(R) such that σ(U) = U implies that δ(U) ⊆ U (Lemma 2.6 of Bhat [3]). Question Let R be a Noetherian ring which is also an algebra over Q. Let σ be an automorphism of R and δ a σ-derivation of R. If O(R) = R[x;σ,δ] is a Noetherian σ(∗)- ring. Is R is a σ(∗)-ring? References [1] V. K. Bhat. Associated prime ideals of skew polynomial rings, Beitrage zur Algebra und Geometrie, Vol. 49(1). 277-283. 2008. [2] V. K. Bhat. On Near Pseudo-valuation rings and their extensions, International Elec- tronic Journal of Algebra, 5:70-77, 2009. [3] V. K. Bhat. Transparent rings and their extensions, New York Journal of Mathematics, Vol. 15. 291-299. 2009. REFERENCES 394 [4] V. K. Bhat. A note on completely prime ideals of Ore extensions, International Jour- nal of Algebra and Computation, Vol. 20(3). 457-463. 2010. [5] V. K. Bhat. Associated prime ideals of weak σ-rigid rings and their extensions, Alge- bra and Discrete Mathematics, Vol.10(1). 8-17. 2010. [6] V. K. Bhat. Prime Ideals of σ(∗)-Rings and their Extensions, Lobachevskii Journal of Mathematics, Vol. 32(1). 102-106. 2011. [7] K. R. Goodearl and R. B. Warfield Jr. An introduction to non-commutative Noethe- rian rings, Cambridge University Press, 1989. [8] J. Krempa. Some examples of reduced rings, Algebra Colloqium, Vol. 3(4). 289-300. 1996. [9] T. K. Kwak. Prime radicals of skew-polynomial rings, International Journal of Math- ematical Sciences, Vol. 2(2). 219-227. 2003. [10] G. Marks. On 2-primal Ore extensions, Communications in Algebra, Vol. 29(5). 2113-2123. 2001. [11] N. H. 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