Associated prime ideals of weak σ-rigid rings and their extensions

Let R be a right Noetherian ring which is also an algebra over Q (Q the field of rational numbers). Let σ be an automorphism of R and δ a σ-derivation of R. Let further σ be such that aσ(a)∈N(R) implies that a∈N(R) for a∈R, where N(R) is the set of nilpotent elements of R. In this paper we...

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Дата:	2010
Автор:	Bhat, V.K.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут прикладної математики і механіки НАН України 2010
Назва видання:	Algebra and Discrete Mathematics
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/154506
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Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	Associated prime ideals of weak σ-rigid rings and their extensions / V.K. Bhat // Algebra and Discrete Mathematics. — 2010. — Vol. 10, № 1. — С. 8–17. — Бібліогр.: 15 назв. — англ.

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spelling	irk-123456789-1545062019-06-16T01:32:44Z Associated prime ideals of weak σ-rigid rings and their extensions Bhat, V.K. Let R be a right Noetherian ring which is also an algebra over Q (Q the field of rational numbers). Let σ be an automorphism of R and δ a σ-derivation of R. Let further σ be such that aσ(a)∈N(R) implies that a∈N(R) for a∈R, where N(R) is the set of nilpotent elements of R. In this paper we study the associated prime ideals of Ore extension R[x;σ,δ] and we prove the following in this direction: Let R be a semiprime right Noetherian ring which is also an algebra over Q. Let σ and δ be as above. Then P is an associated prime ideal of R[x;σ,δ] (viewed as a right module over itself) if and only if there exists an associated prime ideal U of R with σ(U)=U and δ(U)⊆U and P=U[x;σ,δ]. We also prove that if R be a right Noetherian ring which is also an algebra over Q, σ and δ as usual such that σ(δ(a))=δ(σ(a)) for all a∈R and σ(U)=U for all associated prime ideals U of R (viewed as a right module over itself), then P is an associated prime ideal of R[x;σ,δ] (viewed as a right module over itself) if and only if there exists an associated prime ideal U of R such that (P∩R)[x;σ,δ]=P and P∩R=U. 2010 Article Associated prime ideals of weak σ-rigid rings and their extensions / V.K. Bhat // Algebra and Discrete Mathematics. — 2010. — Vol. 10, № 1. — С. 8–17. — Бібліогр.: 15 назв. — англ. 1726-3255 2000 Mathematics Subject Classification:16-XX; 16N40, 16P40, 16S36. http://dspace.nbuv.gov.ua/handle/123456789/154506 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	Let R be a right Noetherian ring which is also an algebra over Q (Q the field of rational numbers). Let σ be an automorphism of R and δ a σ-derivation of R. Let further σ be such that aσ(a)∈N(R) implies that a∈N(R) for a∈R, where N(R) is the set of nilpotent elements of R. In this paper we study the associated prime ideals of Ore extension R[x;σ,δ] and we prove the following in this direction: Let R be a semiprime right Noetherian ring which is also an algebra over Q. Let σ and δ be as above. Then P is an associated prime ideal of R[x;σ,δ] (viewed as a right module over itself) if and only if there exists an associated prime ideal U of R with σ(U)=U and δ(U)⊆U and P=U[x;σ,δ]. We also prove that if R be a right Noetherian ring which is also an algebra over Q, σ and δ as usual such that σ(δ(a))=δ(σ(a)) for all a∈R and σ(U)=U for all associated prime ideals U of R (viewed as a right module over itself), then P is an associated prime ideal of R[x;σ,δ] (viewed as a right module over itself) if and only if there exists an associated prime ideal U of R such that (P∩R)[x;σ,δ]=P and P∩R=U.
format	Article
author	Bhat, V.K.
spellingShingle	Bhat, V.K. Associated prime ideals of weak σ-rigid rings and their extensions Algebra and Discrete Mathematics
author_facet	Bhat, V.K.
author_sort	Bhat, V.K.
title	Associated prime ideals of weak σ-rigid rings and their extensions
title_short	Associated prime ideals of weak σ-rigid rings and their extensions
title_full	Associated prime ideals of weak σ-rigid rings and their extensions
title_fullStr	Associated prime ideals of weak σ-rigid rings and their extensions
title_full_unstemmed	Associated prime ideals of weak σ-rigid rings and their extensions
title_sort	associated prime ideals of weak σ-rigid rings and their extensions
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2010
url	http://dspace.nbuv.gov.ua/handle/123456789/154506
citation_txt	Associated prime ideals of weak σ-rigid rings and their extensions / V.K. Bhat // Algebra and Discrete Mathematics. — 2010. — Vol. 10, № 1. — С. 8–17. — Бібліогр.: 15 назв. — англ.
series	Algebra and Discrete Mathematics
work_keys_str_mv	AT bhatvk associatedprimeidealsofweaksrigidringsandtheirextensions
first_indexed	2025-07-14T06:35:53Z
last_indexed	2025-07-14T06:35:53Z
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fulltext	Algebra and Discrete Mathematics RESEARCH ARTICLE Volume 10 (2010). Number 1. pp. 8 – 17 c© Journal “Algebra and Discrete Mathematics” Associated prime ideals of weak σ-rigid rings and their extensions V. K. Bhat Communicated by M. Ya. Komarnytskyj Abstract. Let R be a right Noetherian ring which is also an algebra over Q (Q the field of rational numbers). Let σ be an automorphism of R and δ a σ-derivation of R. Let further σ be such that aσ(a) ∈ N(R) implies that a ∈ N(R) for a ∈ R, where N(R) is the set of nilpotent elements of R. In this paper we study the associated prime ideals of Ore extension R[x;σ, δ] and we prove the following in this direction: Let R be a semiprime right Noetherian ring which is also an algebra over Q. Let σ and δ be as above. Then P is an associated prime ideal of R[x;σ, δ] (viewed as a right module over itself) if and only if there exists an associated prime ideal U of R with σ(U) = U and δ(U) ⊆ U and P = U [x;σ, δ]. We also prove that if R be a right Noetherian ring which is also an algebra over Q, σ and δ as usual such that σ(δ(a)) = δ(σ(a)) for all a ∈ R and σ(U) = U for all associated prime ideals U of R (viewed as a right module over itself), then P is an associated prime ideal of R[x;σ, δ] (viewed as a right module over itself) if and only if there exists an associated prime ideal U of R such that (P ∩R)[x;σ, δ] = P and P ∩R = U . 1. Introduction and preliminaries Notation: All rings are associative with identity. Throughout this paper R denotes a ring with identity 1 6= 0. The prime radical of R is denoted by P (R). The set of nilpotent elements of R is denoted by N(R). The fields The author would like to express his sincere thanks to the referee for comments and suggestions (regarding inclusion of more examples) to give the paper the present shape. 2000 Mathematics Subject Classification: 16-XX; 16N40, 16P40, 16S36. Key words and phrases: Ore extension, automorphism, derivation, associated prime. V. K. Bhat 9 of rational numbers, real numbers and complex numbers are denoted by Q, R, C respectively. For any subset J of a right R-module M , annihilator of J is denoted by Ann(J). The set of prime ideals of R is denoted by Spec(R), the set of associated prime ideals of R (viewed as a right module over itself) is denoted by Ass(RR), and the set of minimal prime ideals of R is denoted by Min.Spec(R). Let R be a right Noetherian ring. For any uniform right R-module J , the assassinator of J is denoted by Assas(J). Let M be a right R-module. Consider the set {Assas(J) \| J is a uniform right R-submodule of M}. We denote this set by A(MR). Remark 1.1. If R is viewed as a right module over itself, we note that Ass(RR) = A(RR) (5Y of Goodearl and Warfield [8]). For any two ideals I, J of R; I ⊂ J means that I is strictly contained in J . Let K be an ideal of a ring R such that σm(K) = K for some integer m ≥ 1, we denote ∩m i=1σ i(K) by K0. Ore extensions: Let R be a ring, σ an endomorphism of R and δ a σ-derivation of R (δ : R → R is an additive map with δ(ab) = δ(a)σ(b) + aδ(b), for all a, b ∈ R). For example let σ be an endomorphism 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 ring homomorphism. Then δ is a σ-derivation of R. We denote the Ore extension R[x;σ, δ] by O(R). If I is an ideal of R such that I is σ-stable; i.e. σ(I) = I and I is δ-invariant; i.e. δ(I) ⊆ I, then we denote I[x;σ, δ] by O(I). We would like to mention that R[x;σ, δ] is the usual set of polynomials with coefficients in R, i.e. { ∑n i=0 x iai, ai ∈ R} in which multiplication is subject to the relation ax = xσ(a) + δ(a) for all a ∈ R. We take coefficients of the polynomials on the right as followed in McConnell and Robson [13]. In case δ is the zero map, we denote the skew polynomial ring R[x;σ] by S(R) and for any ideal I of R with σ(I) = I, we denote I[x;σ] by S(I). In case σ is the identity map, we denote the differential operator ring R[x; δ] by D(R) and for any ideal J of R with δ(J) ⊆ J , we denote J [x; δ] by D(J). 10 Associated Prime ideals of weak σ-rigid rings Ore-extensions (skew-polynomial rings and differential operator rings) have been of interest to many authors. For example see [1, 4, 5, 7, 8, 10, 12, 13]. Prime ideals: This article concerns the study of prime ideals of Ore extensions (skew polynomial rings). Regarding associated prime ideals of Ore extension R[x;σ, δ], we have the following from S. Annin [1]: Definition (2.1) of Annin [1]: Let R be a ring and MR be a right R-module. Let σ be an endomorphism of R and δ be a σ-derivation of R. MR is said to be σ-compatible if for each m ∈ M , r ∈ R, we have mr = 0 if and only if mσ(r) = 0. Moreover MR is said to be δ-compatible if for each m ∈ M , r ∈ R, we have mr = 0 implies mδ(r) = 0. If MR is both σ-compatible and δ-compatible, MR is said to be (σ − δ)-compatible. Theorem (2.3) of Annin [1]: Let R be a ring. Let σ be an endo- morphism of R and δ a σ-derivation of R and MR be a right R-module. If MR is (σ − δ)-compatible, then Ass(M [x]S) = {P [x] \| P ∈ Ass(MR)}. In [12], Leroy and Matczuk have investigated the relationship between the associated prime ideals of an R-module MR and that of the induced S-module MS , where S = R[x;σ, δ] (σ is an automorphism and δ is a σ-derivation of a ring R). They have proved the following: Theorem (5.7) of [12]: Suppose MR contains enough prime sub- modules and let for Q ∈ Ass(MS). If for every P ∈ Ass(MR), σ(P ) = P , then Q = PS for some P ∈ Ass(MR). Motivated by these developments, I investigated the nature of associ- ated prime ideals of R[x;σ, δ] over a right Noetherian ring R and their relation with those of the coefficient ring R. In this way I generalized Theorem (2.4) and Theorem (3.7) of Bhat [4] for associated prime ideals case. The minimal prime ideal case has been generalized in Lemma (2.2) of Bhat [5]. Before we state these known results we require the following notation: Let R be a right Noetherian ring. We know that Ass(RR) is finite and σj(U) ∈ Ass(RR) for any U ∈ Ass(RR), and for all integers j ≥ 1, therefore, there exists an integer m ≥ 1 such that σm(U) = U for all U ∈ Ass(RR). We denote ∩m i=1σ i(U) by U0 as mentioned in the introduction. Since Min.Spec(R) is also finite, same notation for Min.Spec(R) also. Theorem (2.4) of [4]: Let R be a right Noetherian ring and σ be an automorphism of R. Then: 1. P ∈ Ass(S(R)S(R)) if and only if there exists U ∈ Ass(RR) such that S(P ∩R) = P and (P ∩R) = U0. V. K. Bhat 11 2. P ∈ Min.Spec(S(R)) if and only if there exists U ∈ Min.Spec(R) Such that S(P ∩R) = P and P ∩R = U0. Theorem (3.7) of [4]: Let R be a right Noetherian Q-algebra and δ be a derivation of R. Then: 1. P ∈ Ass(D(R)D(R)) if and only if P = D(P ∩ R) and P ∩ R ∈ Ass(RR). 2. P ∈ Min.Spec(D(R)) if and only if P = D(P ∩ R) and P ∩ R ∈ Min.Spec(R). Before we state the main result, we require the following: Weak σ-rigid rings: Let R be a ring and σ be an endomorphism of R. Recall that in [11], σ is called a rigid endomorphism if aσ(a) = 0 implies a = 0 for a ∈ R, and R is called a σ-rigid ring. Example 1.2. Let R = C, and σ : R → R be the map defined by σ(a+ ib) = a− ib, a, b ∈ R. Then it can be seen that R is a σ-rigid ring. Definition 1.3. (Ouyang [14]): Let R be a ring and σ be an endomor- phism of R. Then R is said to be a weak σ-rigid ring if aσ(a) ∈ N(R) if and only if a ∈ N(R) for a ∈ R. Example 1.4. (Example (2.1) of Ouyang [14]: Let σ be an endomorphism of a ring R such that R is a σ-rigid ring. Let A = {   a b c 0 a d 0 0 a   \| a, b, c, d ∈ R } be a subring of T3(R), the ring of upper triangular matrices over R. Now σ can be extended to an endomorphism σ of A by σ((aij)) = (σ(aij)). The it can be seen that A is a weak σ-rigid ring. 2. Main results We now state the main result in the form of the following Theorem: Theorem A: Let R be a semiprime right Noetherian ring, which is also an algebra over Q. Let σ be an automorphism of R such that R is a weak σ-rigid ring and δ be a σ-derivation of R. Then P ∈ Ass(O(R)O(R)) if and only if there exists U ∈ Ass(RR) such that O(P ∩ R) = P and (P ∩R) = U . This result has been proved in Theorem (2.6). Towards the proof of the above Theorem, we require the following: 12 Associated Prime ideals of weak σ-rigid rings Recall that an ideal of a ring R is said to be completely semiprime if a2 ∈ R implies that a ∈ R. Let R be a Noetherian ring and σ an automorphism of R. We now give a necessary and sufficient condition for R to be a weak σ-rigid ring in the following Theorem: Theorem 2.1. Let R be a Noetherian ring. Let σ be an automorphism of R. Then R is a weak σ-rigid ring if and only if N(R) is completely semiprime. Proof. First of all we show that σ(N(R)) = N(R). We have σ(N(R)) ⊆ N(R) as σ(N(R)) is a nilpotent ideal of R. Now for any n ∈ N(R), there exists a ∈ R such that n = σ(a). So I = σ−1(N(R)) = {a ∈ R such that σ(a) = n ∈ N(R)} is an ideal of R. Now I is nilpotent, therefore I ⊆ N(R), which implies that N(R) ⊆ σ(N(R)). Hence σ(N(R)) = N(R). Now let R be a weak σ-rigid ring. We will show that N(R) is completely semiprime. Let a ∈ R be such that a2 ∈ N(R). Then aσ(a)σ(aσ(a)) = aσ(a)σ(a)σ2(a) ∈ σ(N(R)) = N(R). Therefore aσ(a) ∈ N(R) and hence a ∈ N(R). So N(R) is completely semiprime. Conversely let N(R) be completely semiprime. We will show that R is a weak σ-rigid ring. Let a ∈ R be such that aσ(a) ∈ N(R). Now aσ(a)σ−1(aσ(a)) ∈ N(R) implies that a2 ∈ N(R), and so a ∈ N(R). Hence R is a weak σ-rigid ring. 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. We note that a reduced is 2-primal and a commutative ring is also 2-primal. For further details on 2-primal rings, we refer the reader to [3, 9]. Proposition 2.2. Let R be a 2-primal right Noetherian ring which is also an algebra over Q. Let σ be an automorphism of R such that R is a weak σ-rigid ring and δ a σ-derivation of R. Then σ(U) = U and δ(U) ⊆ U for all U ∈ Min.Spec(R). Proof. Let R be 2-primal weak σ-rigid ring. Then N(R) = P (R) and aσ(a) ∈ N(R) implies that a ∈ N(R). Therefore, aσ(a) ∈ P (R) implies that a ∈ P (R). We will now show that P (R) is completely semiprime. 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). We next show that σ(U) = U for all U ∈ Min.Spec(R). Let U = U1 be a minimal prime ideal of R. 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 V. K. Bhat 13 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). Now P(R) is completely semiprime implies that 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 . Let now V = {a ∈ U \| such that δk(a) ∈ U for all integers k ≥ 1}. First of all, we will show that V is an ideal of R. Let a, b ∈ V . Then δk(a) ∈ U and δk(b) ∈ U for all integers k ≥ 1}. Now δk(a − b) = δk(a)− δk(b) ∈ U for all k ≥ 1}. Therefore a− b ∈ V . Also it is easy to see that for any a ∈ V and for any r ∈ R, ar ∈ V and ra ∈ V . Therefore V is a δ-invariant ideal of R. We will now show that V ∈ Spec(R). Suppose V /∈ Spec(R). Let a /∈ V , b /∈ V be such that aRb ⊆ V . Let t, s be least such that δt(a) /∈ U and δs(b) /∈ U . Now there exists c ∈ R such that δt(a)cσt(δs(b)) /∈ U . Let d = σ−t(c). Now δt+s(adb) ∈ U as aRb ⊆ V . This implies on simplification that δt(a)σt(d)σt(δs(b)) + u ∈ U , where u is sum of terms involving δl(a) or δm(b), where l < t and m < s. Therefore by assumption u ∈ U which implies that δt(a)σt(d)σt(δs(b)) ∈ U . This is a contradiction. Therefore, our supposition must be wrong. Hence V ∈ Spec(R). Now V ⊆ U , so V = U as U ∈ Min.Spec(R). Hence δ(U) ⊆ U . Corollary 2.3. Let R be a 2-primal right Noetherian ring which is also an algebra over Q. Let σ be an automorphism of R such that σ(U) = U for all U ∈ Min.Spec(R). Let δ be a σ-derivation of R. Then δ(U) ⊆ U . Lemma 2.4. Let R be a right Noetherian ring which is also an algebra over Q. Let σ be an automorphism of R such that R is a weak σ-rigid ring and δ a σ-derivation of R. Then 1. If U is a minimal prime ideal of R, then O(U) is a minimal prime ideal of of O(R) and O(U) ∩R = U . 2. If P is a minimal prime ideal of O(R), then P ∩ R is a minimal prime ideal of R. Proof. (1) Let U be a minimal prime ideal of R. Then by Proposition (2.2) σ(U) = U and δ(U) ⊆ U . Now on the same lines as in Theorem (2.22) of Goodearl and Warfield [8] we have O(U) ∈ Spec(O(R)). Suppose L ⊂ O(U) be a minimal prime ideal of O(R). Then L ∩R ⊂ U is a prime ideal of R, a contradiction. Therefore O(U) ∈ Min.Spec(O(R)). Now it is easy to see that O(U) ∩R = U . (2) We note that x /∈ P for any prime ideal P of O(R) as it is not a zero divisor. Now the proof follows on the same lines as in Theorem (2.22) of Goodearl and Warfield [8] using Lemma (2.1) and Lemma (2.2) of Bhat [2] and Proposition (2.2). 14 Associated Prime ideals of weak σ-rigid rings Theorem 2.5 (Hilbert Basis Theorem). Let R be a right/left Noetherian ring. Let σ and δ be as usual. Then the ore extension O(R) = R[x;σ, δ] is right/left Noetherian. Proof. See Theorem (2.6) of Goodearl and Warfield [8]. With this we now state and prove Theorem A: Theorem 2.6. Let R be a semiprime right Noetherian ring, which is also an algebra over Q. Let σ be an automorphism of R such that R is a weak σ-rigid ring and δ be a σ-derivation of R. Then P ∈ Ass(O(R)O(R)) if and only if there exists U ∈ Ass(RR) such that O(P ∩ R) = P and P ∩R = U . Proof. O(R) is right Noetherian by Theorem (2.5). Let P ∈ Ass(O(R)O(R)). Now by Remark (1.1) Ass(O(R)O(R)) = A(O(R)(R)). Let P = Ann(I) = Assas(I) for some ideal I of O(R) such that I is uniform as a right O(R)-module. Choose f ∈ I to be nonzero of minimal degree (with lead- ing coefficient an). Let U = Ann(anR) = Assas(anR). Now R is right Noetherian implies that Ass(RR) = A(RR), and since R is semiprime, U ∈ Min.Spec(R) by Proposition (2.2.14) of McConnell and Robson [13]. Now R is a weak σ-rigid ring, therefore, Proposition (2.2) implies that σ(U) = U and δ(U) ⊆ U . So O(U) is an ideal of O(R). Now fU = 0. Therefore fO(R)U ⊆ fUO(R) = 0, i.e. U ⊆ P ∩ R. But it is clear that P ∩R ⊆ U . Thus P ∩R = U . Conversely let U = Ann(cR) = Assas(cR), c ∈ R. Now R is right Noetherian implies that Ass(RR) = A(RR), and since R is semiprime, U ∈ Min.Spec(R) by Proposition (2.2.14) of McConnell and Robson [13]. Now R is a weak σ-rigid ring, therefore, Proposition (2.2) implies that σ(U) = U and δ(U) ⊆ U . Now it can be easily seen that O(U) = Ann(chO(R)) for all h ∈ O(R). Therefore O(U) = Ann(cO(R)) = Assas(cO(R)). Example 2.7. 1. R as in Example 1.2 is a semiprime weak σ-rigid ring, but R being a field has no ideals and is therefore a trivial example. 2. Let τ be the conjugacy map on C. Let R = { ( a b 0 a ) \| a, b ∈ C } . Define σ : R → R by σ((aij)) = (τ(aij)). Then it can be seen that σ is an endomorphism of R and R is a weak σ-rigid ring. Now for any s ∈ R, define δs : R → R by δs(a) = as − sσ(a), for a ∈ R. Then δs is a σ-derivation of R. Let V. K. Bhat 15 U = { ( a b 0 0 ) ∣ ∣ ∣ a, b ∈ C } ∈ Ass(RR). In fact U = Ann(I) = Assas(I), where I = { ( 0 0 0 c ) ∣ ∣ ∣ c ∈ C } is a right ideal of R. Now we note that σ(I) = I, δs(I) ⊆ I, Then it can be seen that σ is an endomorphism of R and σ(U) ⊆ U . and δs(U) ⊆ U . Also O(U) ∈ Ass(O(R)O(R)). In fact O(U) = Ann(O(I)) = Assas(O(I)). Example 2.8. Now let R = F × F , F a field and σ : R → R defined by σ((u, v)) = (v, u) for u, v ∈ F . Then σ is an automorphism of R. But R is not a weak σ-rigid ring as for any 0 6= a ∈ F , we have (a, 0)σ((a, 0)) = (0, 0) ∈ N(R), but (a, 0) /∈ N(R). Proposition 2.9. Let R be a Noetherian Q-algebra. 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. Then U ∈ Min.Spec(R) with σ(U) = U implies that δ(U) ⊆ U . Proof. See Lemma (2.6) of Bhat [6]. We now prove the following Theorem: Theorem 2.10. Let R be a right Noetherian ring which is also an algebra over Q, σ be an automorphism of R and δ a σ-derivation of R such that σ(δ(a)) = δ(σ(a)) for all a ∈ R and σ(U) = U for all U ∈ A(RR). Then P ∈ Ass(O(R)O(R)) if and only if there exists U ∈ Ass(RR) such that O(P ∩R) = P and P ∩R = U . Proof. O(R) is right Noetherian by Theorem (2.5). Let J ∈ Ass(O(R)O(R)). Now by Remark (1.1) Ass(O(R)O(R)) = A(O(R)(R)). Let P = Ann(I) = Assas(I) for some ideal I of O(R) such that I is uniform as a right O(R)- module. Choose f ∈ I to be nonzero of minimal degree (with leading coefficient an). Let U = Ann(anR) = Assas(anR). Now R is right Noethe- rian implies that Ass(RR) = A(RR). Now by hypothesis σ(U) = U , and therefore, Proposition (2.9) implies that δ(U) ⊆ U . So O(U) is an ideal of O(R). Now fU = 0. Therefore fO(R)U ⊆ fUO(R) = 0. So U ⊆ P ∩ R. But it is clear that P ∩R ⊆ U . Thus P ∩R = U . Conversely let U = Ann(cR) = Assas(cR), c ∈ R. Now R is right Noetherian implies that Ass(RR) = A(RR). Now by hypothesis σ(U) = U , and therefore, Proposition (2.9) implies that δ(U) ⊆ U . Now it can be easily seen that O(U) = Ann(chO(R)) for all h ∈ O(R). Therefore O(U) = Ann(cO(R)) = Assas(cO(R)). Example 2.11. Let R = { ( a b 0 a ) ∣ ∣ ∣ a, b ∈ R } . Then U = { ( a b 0 0 ) ∣ ∣ ∣ a, b ∈ R } ∈ Ass(RR). 16 Associated Prime ideals of weak σ-rigid rings In fact U = Ann(I) = Assas(I), where I = { ( 0 0 0 c ) ∣ ∣ ∣ c ∈ R } is a right ideal of R. Let σ : R → R be defined by σ ( ( a b 0 a ) ) = ( a 0 0 a ) . Then it can be seen that σ is an endomorphism of R and σ(U) ⊆ U . For any s ∈ R, define δs : R → R by δs(a) = as− sσ(a), for a ∈ R. Then δs is a σ-derivation of R. Also we see that σ(δs(u)) = δs(σ(u)) for all u ∈ R. For let u = ( a b 0 a ) and s = ( p q 0 p ) . Then σ(δs(u)) = ( 0 0 0 0 ) and δs(σ(u)) = ( 0 0 0 0 ) . Now we note that σ(I) = I, δs(I) ⊆ I and δs(U) ⊆ U . Also O(U) ∈ Ass(O(R)O(R)). In fact O(U) = Ann(O(I)) = Assas(O(I)). Example 2.12. Let R = ( R R 0 R ) . Then P = ( R R 0 0 ) ∈ Ass(RR). In fact P = Ann(I) where I = ( 0 0 0 R ) is a right ideal of R. 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 σ(P ) ⊆ P . For any s ∈ R, define δs : R → R by δs(a) = as− sσ(a), for a ∈ R. Then δs is a σ-derivation of R. But we see that σ(δs(u)) 6= δs(σ(u)) for all u ∈ R. Let u = ( a b 0 c ) and s = ( p q 0 r ) . Then σ(δs(u)) = ( 0 pb+ qc− aq 0 0 ) and δs(σ(u)) = ( 0 0 0 0 ) . Example 2.13. 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 (a, b) ∈ R, σ(δr((a, b))) = 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. We see that U = 0×R ∈ Ass(RR). In fact U = Ann(R×{0}) = Assas(R×{0}). But we note that σ(U) 6= U . V. K. Bhat 17 References [1] S. Annin, Associated primes over Ore extension rings, J. Algebra Appl. 3 (2004), no. 2, 193-205. [2] V. K. Bhat, Polynomial rings over pseudovaluation rings, Int. J. Math. and Math. Sc. 2007 (2007), Art. ID 20138. [3] V. K. Bhat, On 2-primal Ore extensions, Ukr. Math. Bull., Vol. 4(2) (2007), 173-179. [4] V. K. Bhat, Associated prime ideals of skew polynomial rings, Beitrдge Algebra Geom., Vol. 49(1) (2008), 277-283. [5] V. K. Bhat, On near-pseudo-valuation rings and their extensions, Int. Electron. J. Algebra, Vol. 5 (2009), 70-77. [6] V. K. Bhat, Transparent rings and their extensions, New York J. Math., Vol. 15 (2009), 291-299. [7] W. D. Blair and L. W. Small, Embedding differential and skew-polynomial rings into artinain rings, Proc. Amer. Math. Soc. 109(4) 1990, 881-886. [8] K. R. Goodearl and R. B. Warfield Jr., An introduction to non-commutative Noetherian rings, Second Edition, Cambridge Uni. Press, 2004. [9] N. K. Kim and T. K. Kwak, Minimal prime ideals in 2-primal rings, Math. Japon., Vol. 50(3) (1999), 415-420. [10] T. K. Kwak, Prime radicals of skew-polynomial rings, Int. J. Math. Sci., Vol. 2(2) (2003), 219-227. [11] J. Krempa, Some examples of reduced rings, Algebra Colloq., Vol. 3(4) (1996), 289-300. [12] A. Leroy and J. Matczuk, On induced modules over Ore extensions, Comm. Algebra 32 (2004), no. 7, 2743-2766. [13] J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, Wiley 1987; revised edition: Amer. Math. Soc. 2001. [14] L. Ouyang, Extensions of generalized α-rigid rings, Int. Electron. J. Algebra, Vol. 3 (2008), 103-116. [15] A. Seidenberg, Differential ideals in rings of finitely generated Type, Amer. J. Math. 89 (1967), 22-42. Contact information V. K. Bhat School of Mathematics, SMVD University, P/o SMVD University, Katra, J and K, India- 182320 E-Mail: vijaykumarbhat2000@yahoo.com Received by the editors: 16.10.2009 and in final form 16.10.2009.

Associated prime ideals of weak σ-rigid rings and their extensions

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