/compile/output.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 8, No. 4, 2015, 462-468 ISSN 1307-5543 – www.ejpam.com Ore Extensions Over (σ,δ)-Rings M. Abrol, V. K. Bhat∗ 1 School of Mathematics, SMVD University, P/O SMVD University, Katra, J and K, India- 182320 Abstract. Let R be a Noetherian, integral domain which is also an algebra over Q (Q is the field of rational numbers). Let σ be an automorphism of R and δ a σ-derivation of R. A ring R is called a (σ,δ)-ring if a(σ(a)+δ(a)) ∈ P(R) implies that a ∈ P(R) for a ∈ R, where P(R) is the prime radical of R. We prove that R is 2-primal if δ(P(R)) ⊆ P(R). We also study the property of minimal prime ideals of R and prove the following in this direction: Let R be a Noetherian, integral domain which is also an algebra over Q. Let σ be an automorphism of R and δ a σ-derivation of R such that R is a (σ,δ)-ring. If P ∈ Min.Spec(R) is such that σ(P) = P, then δ(P) ⊆ P. Further if δ(P(R)) ⊆ P(R), then P[x;σ,δ] is a completely prime ideal of R[x;σ,δ]. 2010 Mathematics Subject Classifications: 16-XX, 16W20, 16P40, 16S50 Key Words and Phrases: Noetherian ring, Ore extension, endomorphism, automorphism, minimal prime ideals, (σ,δ)-rings and 2-primal. 1. Introduction and Preliminaries All rings are associative with identity 1 6= 0, unless otherwise stated. The prime radical and the set of nilpotent elements of R are denoted by P(R) and N(R) respectively. The ring of integers is denoted by Z and the field of rational numbers by Q, unless otherwise stated. The set of minimal prime ideals of R is denoted by Min.Spec(R). We begin with the following: Definition 1. Let R be a ring, σ an endomorphism of R and δ a σ-derivation of R, which is defined as an additive map from R to R such that [12] δ(ab) = δ(a)σ(b) + aδ(b), for all a, b ∈ R. Example 1. Let R= Z[ p 2]. Then σ : R→ R defined as σ(a+ b p 2) = a− b p 2 for a+ b p 2 ∈ R ∗Corresponding author. Email address: vijaykumarbhat2000@yahoo.com (V. Bhat) http://www.ejpam.com 462 c© 2015 EJPAM All rights reserved. M. Abrol, V. Bhat / Eur. J. Pure Appl. Math, 8 (2015), 462-468 463 is an endomorphism of R. For any s ∈ R, Define δs : R→ R by δs(a+ b p 2) = (a+ b p 2)s− sσ(a+ b p 2) for a+ b p 2 ∈ R. Then δs is a σ-derivation of R. Recall that R[x;σ,δ] is the usual polynomial ring with coefficients in R where multiplica- tion 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 the Ore extension R[x;σ,δ] by O(R). An ideal I of a ring R is called σ-stable if σ(I) = I and is called δ-invariant if δ(I) ⊆ I . If an ideal I of R is σ-stable and δ-invariant, then I[x;σ,δ] is an ideal of O(R) and as usual we denote it by O(I). Definition 2. A completely prime ideal in a ring R is any ideal such that R/P is a domain [7]. Also an ideal P of a ring R is said to be completely prime if ab ∈ P implies that a ∈ P or b ∈ P for a, b ∈ R. In commutative sense completely prime and prime have the same meaning. We also note that a completely prime ideal of a ring R is a prime ideal, but the converse need not be true. The following example shows that a prime ideal need not be a completely prime ideal. Example 2 (Example 1.1 of [2]). 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 completely 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. There are examples of rings (non-commutative) in which prime ideals are completely prime. Example 3 (Example 1.2 of [2]). Let R = � Z Z 0 Z � . Then P1 = � Z Z 0 0 � , P2 = � 0 Z 0 Z � and P3 = � 0 Z 0 0 � are prime ideals of R. Now all these are completely prime also. Definition 3. A minimal prime ideal in a ring R is any prime ideal of R that does not properly contain any other prime ideal [3]. Example 4. In example 1.2 of [2] (discussed above), P3 = � 0 Z 0 0 � is minimal prime ideal. Further more there are examples of rings in which minimal prime ideals are completely prime. For example a reduced ring. If R is a prime ring, then 0 is a minimal prime ideal of R and it is the only one. In Proposition (3.3) of [6], it has been shown that any prime ideal U in a ring R contains a minimal prime ideal. Further it has been proved that there exists M. Abrol, V. Bhat / Eur. J. Pure Appl. Math, 8 (2015), 462-468 464 only finitely many minimal prime ideals in a Noetherian ring R and there is a finite product of minimal prime ideals (repetition allowed) that equals zero. An example of a ring which has infinitely many minimal prime ideals is: Example 5 (Exercise 3C of [7]). Let X be an infinite set, K a field, and R the ring of all functions from X to K. For x ∈ X , let Px be the set of those functions in R which vanish at x. Then each Px is a minimal prime ideal of R. It is also known that [6] in a right Noetherian ring which is also an algebra over Q, δ a σ-derivation of R and U a minimal prime ideal of R, δ(U) ⊆ U . Definition 4. A ring R is said to be 2-primal if and only if P(R) = N(R) [4]. Example 6. Let R= (Z/8Z⊕Z/8Z). Then R is a commutative ring and hence 2-primal. Also a reduced ring is 2-primal and so is a commutative Noetherian ring. Part of the attraction of 2-primal rings in addition to their being a common generalization of commutative rings and rings without nilpotent elements lies in the structure of their prime ideals. We refer to [4, 5, 8, 9, 11, 13, 14] for more details on 2-primal rings. Definition 5. Let R be a ring and σ an endomorphism of R. Then R is said to be σ(∗)-ring if aσ(a) ∈ P(R) implies that a ∈ P(R) for a ∈ R [3]. We note that if R is a Noetherian ring and σ 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 [Theorem (2.3) of 3]. Definition 6. Let R be a ring. Let σ be an automorphism of R and δ a σ-derivation of R. Then R is a δ-ring if aδ(a) ∈ P(R) implies that a ∈ P(R) for a ∈ P(R) [1]. Note that a ring with identity is not a δ-ring as 1δ(1) = 0, but 1 6= 0. Also from [1] we know that if R is a δ-Noetherian Q-algebra such that σ(δ(a)) = δ(σ(a)), for all a ∈ R; σ(P) = P, for all P ∈ Min.Spec(R) and δ(P(R)) ⊆ P(R), then R[x;σ,δ] is 2-primal. We now generalize these notions as follows: Definition 7. Let R be a ring. Let σ be an endomorphism of R and δ a σ-derivation of R. Then R is said to be a (σ,δ)-ring if a(σ(a) +δ(a)) ∈ P(R) implies that a ∈ P(R) for a ∈ R. Example 7. Let R= � Z Z 0 Z � . Then P(R) = � 0 Z 0 0 � . Let σ : R→ R be defined by σ � � a b 0 c � � = � a −b 0 c � , for all a, b, c ∈ Z. Then it can be seen that σ is an endomorphism of R. Define δ : R→ R by δ(a) = a−σ(a), for alla ∈ R. M. Abrol, V. Bhat / Eur. J. Pure Appl. Math, 8 (2015), 462-468 465 Clearly, δ is a σ-derivation of R. Now let A= � a b 0 c � . A[σ(A) +δ(A)] ∈ P(R) implies that � a b 0 c � ¦ σ � � a b 0 c � � + � a b 0 c � −σ � � a b 0 c � �© ∈ P(R) or � a b 0 c � ¦ � a −b 0 c � + � a b 0 c � −σ � � a b 0 c � �© ∈ P(R) which gives on simplification, � a2 ab+ bc 0 c2 � ∈ P(R) = � 0 Z 0 0 � which implies that a2 = 0, c2 = 0, i.e. a = 0, c = 0. Therefore, A= � a b 0 c � = � 0 b 0 0 � ∈ P(R). Hence R is a (σ,δ)-ring. Example 8. Let R = Z2 ⊕ Z2. Then R is a commutative reduced ring. Define an automorphism σ : R→ R by σ((a, b)) = (b, a) for a, b ∈ Z2. Also δ : R→ R defined by δ((a, b)) = (a − b, 0) for a, b ∈ Z2 is a σ-derivation of R. Here P(R) = {0}. But R is not a (σ,δ)-ring, for take (a, b) = (0,1). With this we prove the following: Theorem 1: Let R be a Noetherian, integral domain which is also an algebra over Q. Let σ be an automorphism of R and δ a σ-derivation of R such that R is a (σ,δ)-ring and δ(P(R)) ⊆ P(R). Then R is 2-primal. Theorem 2: Let R be a Noetherian, integral domain which is also an algebra over Q. Let σ be an automorphism of R and δ a σ-derivation of R such that R is a (σ,δ)-ring. If P ∈ Min.Spec(R) is such that σ(P) = P, then δ(P) ⊆ P. Theorem 4: Let R be a Noetherian, integral domain which is also an algebra over Q. Let σ be an automorphism of R and δ a σ-derivation of R such that R is a (σ,δ)-ring and δ(P(R)) ⊆ P(R). Let P ∈ Min.Spec(R) be such that σ(P) = P, then O(P) is a completely prime ideal of O(R). 2. Proof of Main Results We now prove Theorems 1, 2 and 3 as follows: Theorem 1. Let R be a Noetherian, integral domain which is also an algebra over Q. Let σ be an automorphism of R and δ a σ-derivation of R such that R is a (σ,δ)-ring and δ(P(R)) ⊆ P(R). Then R is 2-primal. M. Abrol, V. Bhat / Eur. J. Pure Appl. Math, 8 (2015), 462-468 466 Proof. Define a map ρ : R/P(R)→ R/P(R) by ρ(a+ P(R)) = δ(a) + P(R) for a ∈ R Also define τ : R/P(R)→ R/P(R) by τ(a+ P(R)) = σ(a) + P(R) for a ∈ R. Then τ is an automorphism of R/P(R) and ρ is a τ-derivation of R/P(R). Also a(σ(a) +δ(a)) ∈ P(R) if and only if (a+ P(R))ρ(a+ P(R)) + (a+ P(R))τ(a+ P(R)) = P(R)inR/P(R). Then as in Proposition (5) of [10], R is a reduced ring. Hence it is 2-primal. For the proof of Theorem 2, we need the following: Proposition 1. Let R be a Noetherian ring which is also an algebra over Q. Let δ be a derivation of R. Then δ(P(R)) ⊆ P(R). Proof. See Proposition (1.1) of [1]. Proposition 2. Let R be a 2-primal ring. Let σ be an automorphism of R and δ a σ-derivation of R such that δ(P(R)) ⊆ P(R). If P ∈ Min.Spec(R) is such that σ(P) = P, then δ(P) ⊆ P. Proof. Let P ∈ Min.Spec(R). Now P is a completely prime ideal, therefore, for any a ∈ P there exists b /∈ P such that ab ∈ P(R) by Corollary (1.10) of Shin [13]. Now δ(P(R)) ⊆ P(R), and therefore δ(ab) ⊆ P(R); i.e., δ(a)σ(b) + aδ(b) ∈ P(R) ⊆ P. Now aδ(b) ∈ P implies that δ(a)σ(b) ∈ P. Now σ(P) = P implies that σ(b) /∈ P and since P is completely prime in R, we have δ(a) ∈ P. Hence δ(P) ⊆ P. Theorem 2. Let R be a Noetherian, integral domain which is also an algebra over Q. Let σ be an automorphism of R and δ a σ-derivation of R such that R is a (σ,δ)-ring. If P ∈ Min.Spec(R) is such that σ(P) = P, then δ(P) ⊆ P. Proof. Let P ∈ Min.Spec(R). Then by Proposition 1, δ(P(R)) ⊆ P(R) and by Theorem 1, R is 2-primal. Since σ(P) = P, the result follows by Proposition 2. For the proof of Theorem 4, we need the following: Theorem 3. Let R be a ring. Let σ be an automorphism of R and δ a σ-derivation of R. Then: (i) For any completely prime ideal P of R with σ(P) = P and δ(P) ⊆ P, O(P) is a completely prime ideal of O(R). (ii) For any completely prime ideal U of O(R), U ∩ R is a completely prime ideal of R. Proof. See Theorem (2.4) of [2]. REFERENCES 467 Theorem 4. Let R be a Noetherian, integral domain which is also an algebra over Q. Let σ be an automorphism of R and δ a σ-derivation of R such that R is a (σ,δ)-ring and δ(P(R)) ⊆ P(R). Let P ∈ Min.Spec(R) be such that σ(P) = P, then O(P) is a completely prime ideal of O(R). Proof. R is 2-primal by Theorem 1 and so by Proposition 2, δ(P) ⊆ P and as in proof of Proposition 2 above, P is a completely prime ideal of R. Now use Theorem 3 and the proof is complete. References [1] V. K. Bhat. Differential operator rings over 2-primal rings, Ukranian Mathematical Bulletin, 5(2), 153-158. 2008. [2] V. K. Bhat. A note on completely prime ideals of ore extensions, International Journal of Algebra and Computation, 20(3), 457-463. 2010. [3] V. K. Bhat. Minimal prime ideals of σ(∗)-rings and their extensions, Armenian Journal of Mathematics, 5(2), 98-104. 2013. [4] G. F. Birkenmeier, H. E. Heatherly, and E. K. Lee. Completely prime ideals and associated radicals, In. S. K.Jain, S. T. Rizvi, eds, Proc. Biennal Ohio State - Denison Conference 1992. Singapore-New Jersey-London-Hongkong: World Scientific (Singapore), 102-129. 1993. [5] G. F. Birkenmeier, J. Y. Kim, and J. K. Park. Polynomial extensions of Baer and quasi-Baer rings, Journal of Pure and Applied Algebra, 159(1), 25-41. 2001. [6] P. Gabriel. Representations Des Algebres De Lie Resoulubles (D Apres J. Dixmier). In Semi- naire Bourbaki, 1968-69, pp 1-22, Lecture Notes in Math. No. 179, Berlin 1971 Springer Verlag, 1971. [7] K. R. Goodearl and R. B. Warfield. An introduction to Non-commutative Noetherian rings, Cambridge University Press, 2004. [8] Y. Hirano Some studies on strongly π-regular ring, Mathematical Journal of Okayama University, 20(2), 141-149. 1978. [9] C.Y. Hong and T.K. Kwak. On minimal strongly prime ideals, Communications in Algebra, 28(10), 4868-4878. 2000. [10] C.Y.Hong, N. K. Kim, and T. K. Kwak. Ore extensions of Baer and p.p-rings,Journal of Pure and Applied Algebra, 151(3), 215-226. 2000. [11] N.K. Kim and T.K.Kwak. Minimal prime ideals in 2-primal rings, Mathematica Japonica, 50(3), 415-420. 1999. REFERENCES 468 [12] J. C. McConnell and J. C. Robson. Noncommutative Noetherian Rings, Wiley,1987; revised edition: AMS, 2001. [13] G. Y. Shin. Prime ideals and sheaf representations of a pseudo symmetric ring, Transactions of American Mathematical Society, 184, 43-60. 1973. [14] S. H. Sun. Non-commutative rings in which every prime ideal is contained in a unique maximal ideal, Journal of Pure and Applied Algebra, 76(2), 179-192. 1991.