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International Journal of Mathematics and Mathematical Sciences
Volume 2008, Article ID 835605, 8 pages
http://dx.doi.org/10.1155/2008/835605
Research Article

Prime Ideals and Strongly Prime Ideals of Skew Laurent Polynomial Rings

School of Mathematical Sciences, Shahrood University of Technology, P.O. Box 316-3619995161, Shahrood, Iran

Received 26 February 2008; Accepted 28 April 2008

Academic Editor: Howard Bell

Copyright © 2008 E. Hashemi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We first study connections between -compatible ideals of and related ideals of the skew Laurent polynomials ring , where is an automorphism of . Also we investigate the relationship of and of with the prime radical and the upper nil radical of the skew Laurent polynomial rings. Then by using Jordan's ring, we extend above results to the case where is not surjective.

1. Introduction

Throughout the paper, always denotes an associative ring with unity. We use , , and to denote the prime radical, the upper nil radical, and the set of all nilpotent elements of , respectively.

Recall that for a ring with an injective ring endomorphism , is the Ore extension of . The set is easily seen to be a left Ore subset of , so that one can localize and form the skew Laurent polynomials ring . Elements of are finite sum of elements of the form , where and are nonnegative integers. Multiplication is subject to and for all .

Now we consider Jordan's construction of the ring (see [1], for more details). Let or be the subset of the skew Laurent polynomial ring . For each , . It follows that the set of all such elements forms a subring of with and for and . Note that is actually an automorphism of , given by to , for each and . We have , by way of an isomorphism which maps to . For an -ideal of , put . Hence is -ideal of . The constructions , are inverses, so there is an order-preserving bijection between the sets of -invariant ideals of and -invariant ideals of .

According to Krempa [2], an endomorphism of a ring is called rigid if implies for . is called an -rigid ring [3] if there exists a rigid endomorphism of . Note that any rigid endomorphism of a ring is a monomorphism and -rigid rings are reduced (i.e., has no nonzero nilpotent element) by Hong et al. [3]. Properties of -rigid rings have been studied in Krempa [2], Hirano [4], and Hong et al. [3, 5].

On the other hand, a ring is called 2-primal if (see [6]). Every reduced ring is obviously a 2-primal ring. Moreover, 2 primal rings have been extended to the class of rings which satisfy , but the converse does not hold [7, Example 3.3]. Observe that is a 2-primal ring if and only if , if and only if is a completely semiprime ideal (i.e., implies that for ) of . We refer to [612] for more detail on 2 primal rings.

Recall that a ring is called strongly prime if is prime with no nonzero nil ideals. An ideal of is strongly prime if is a strongly prime ring. All (strongly) prime ideals are taken to be proper. We say an ideal of a ring is minimal (strongly) prime if is minimal among (strongly) prime ideals of . Note that (see [13]) is a minimal strongly prime ideal of .

Recall that an ideal of is completely prime if implies or for . Every completely prime ideal of is strongly prime and every strongly prime ideal is prime.

According to Hong et al. [5], for an endomorphism of a ring , an -ideal is called to be -rigid ideal if implies that for . Hong et al. [5] studied connections between -rigid ideals of and related ideals of some ring extensions. Also they studied relationship of and of with the prime radical and the upper nil radical of the Ore extension of in the cases when either or is an -rigid ideal of and obtaining the following result. Let (resp., ) be an -rigid -ideal of . Then (resp., ).

In [14], the authors defined -compatible rings, which are a generalization of -rigid rings. A ring is called -compatible if for each , . In this case, clearly the endomorphism is injective. In [14, Lemma 2.2], the authors showed that is -rigid if and only if is -compatible and reduced. Thus, the -compatible ring is a generalization of -rigid ring to the more general case where is not assumed to be reduced.

Motivated by the above facts, for an endomorphism of a ring , we define -compatible ideals in which are a generalization of -rigid ideals. For an ideal , we say that is an -compatible ideal of if for each , . The definition is quite natural, in the light of its similarity with the notion of -rigid ideals, where in Proposition 2.3, we will show that is an -rigid ideal if and only if is an -compatible ideal and completely semiprime.

In this paper, we first study connections between -compatible ideals of and related ideals of the skew Laurent polynomial ring , where is an automorphism of . Also we investigate the relationship of and of with the prime radical and the upper nil radical of the skew Laurent polynomials. Then by using Jordan's ring, we extend above results to the case where is not surjective.

2. Prime Ideals and Strongly Prime Ideals of Skew Laurent Polynomial Rings

Recall that an ideal of is called an -ideal if ; is called -invariant if . If is an -ideal, then defined by is an endomorphism. Then we have the following proposition.

Proposition 2.1. Let be an ideal of a ring . Then the following statements are equivalent:
(1) is an -compatible ideal;(2) is -compatible.

Proof. It is clear.

Proposition 2.2. Let be an -compatible ideal of a ring . Then
(1) is -invariant;(2) if , then for every positive integer ; conversely, if or for some positive integer , then .

Proof. This follows from [15, Lemma 2.2 and Proposition 2.3].

Recall from [16] that a one-sided ideal of a ring has the insertion of factors property (or simply, IFP) if implies for (Bell in 1970 introduced this notion for ).

Proposition 2.3 (see [15], Proposition 2.3 2.4). Let be a ring, an ideal of , and an endomorphism of . Then the following conditions are equivalent:
(1) is -rigid ideal of ;(2) is -compatible, semiprime and has the IFP;(3) is -compatible and completely semiprime.

For an -ideal of , put .

Proposition 2.4. (1) If is an -compatible ideal of , then is an -compatible ideal of .
(2) If is an -compatible ideal of , then and is an -compatible ideal of .
(3) If is a completely (semi)prime -compatible ideal of , then is a completely (semi)prime -compatible ideal of .
(4) If is a completely (semi)prime -compatible ideal of , then and is a completely (semi)prime -compatible ideal of .
(5) If is a prime -compatible ideal of , then is a prime -compatible ideal of .

Proof. (1) Since is an -ideal of , is an ideal of . Now, let . Hence and that . Thus , since is -compatible. Consequently . Therefore is -compatible.
(2) Let and . Then for each . Hence for each . Thus , since is -compatible. Therefore . Now, let . Then and that , since is -compatible. Thus . Consequently, .
(3) Let . Then and that . Hence , by Proposition 2.2. Thus or , since is completely prime. Consequently, or .
(4) By (2), and is a -compatible ideal of . Since is completely (semi)prime and , hence is completely (semi)prime. Let . Then , by Proposition 2.2. Hence or , since is prime. Consequently or . Therefore is a prime ideal of .

Theorem 2.5. Let be a strongly prime -compatible ideal of . Then is a strongly prime ideal of .

Proof. Since is a prime -compatible ideal of , hence is a prime ideal of , by Proposition 2.4. We show that is a strongly prime ideal of . Assume is a nil ideal of . Let . Then . Since is a nil ideal, hence for some . Hence for some and . Thus . Hence , since is -compatible. Then is a nil ideal of for each . Hence , for each . Therefore . Consequently, is a strongly prime ideal of .

Note that if is an -ideal of , then is an ideal of the skew Laurent polynomials ring .

Theorem 2.6. Let be an automorphism of . Let be a semiprime -compatible ideal of . Assume and . Then the following statements are equivalent:
(1);(2) for each .

Proof. (1)(2). Assume . ThenHence . Thus , since is -compatible. Next, replacing by , where . Then . Hence and that , since is -compatible. Thus . Hence , since is semiprime. Continuing this process, we obtain , for . Hence from -compatibility of , we get . Using induction on , we obtain for each .
(2)(1). It follows from Proposition 2.2.

Corollary 2.7. Let be an automorphism on . If is a (semi)prime -compatible ideal of , then is a (semi)prime ideal of .

Proof. Assume that is a prime -compatible ideal of . Let and such that . Then for each , by Theorem 2.6. Assume . Hence for some . Thus for each , since is prime. Therefore . Consequently, is a prime ideal of .

Theorem 2.8. If each minimal prime ideal of is -compatible, then .

Proof. Let be a minimal prime ideal of . By Proposition 2.4, is a -compatible ideal of . Assume . Then , since is -compatible. Hence or . Thus or . Therefore is a prime ideal of . Thus is a prime ideal of , by Corollary 2.7. Consequently, .

In [14], the authors give some examples of -compatible rings however they are not -rigid. Note that there exists a ring for which every nonzero proper ideal is -compatible but is not -compatible. For example, let , where is a field, and the endomorphism of is defined by for .

The following examples show that there exists -compatible ideals which are not -rigid.

Example 2.9 (see [2.95], Example 2.5). Let be a field. Let , where is the ring of polynomials over . Then is a subring of the matrix ring over the ring . Let be an automorphism defined by , where is a fixed nonzero element of . Let be an irreducible polynomial in . Let , where is the principal ideal of generated by . Then is an -compatible ideal of but is not -rigid. Indeed, since , but for . Thus is not -rigid.

Example 2.10.10 (see [17], Example 2.10). Let be the field of integers modulo 2 and be the free algebra of polynomials with zero constant term in noncommuting indeterminates over . Note that is a ring without unity. Consider an ideal of , say , generated by , , , , , , , with and with . Then has the IFP. Let be an inner automorphism (i.e., there exists an invertible element such that for each ). Then is -compatible, since has the IFP. But is not -rigid, since is not completely semiprime.

Theorem 2.11. Let be an automorphism of . If each minimal prime ideal of is -compatible, then .

Proof. It follows from Corollary 2.7.

The following example shows that there exists a ring such that all minimal prime ideals are -compatible, but are not -rigid.

Example 2.12.12.12 (see [15], Example 2.12.11). Let be the matrix ring over the ring . Then is the only prime ideal of . Let be the endomorphism defined by . Then is an automorphism of and is -compatible. However, is not -rigid, since , but .

Theorem 2.13. Let be an automorphism of . If is a completely (semi)prime -compatible ideal of , then is a completely (semi)prime ideal of .

Proof. Let be a completely prime ideal of . is domain, hence it is a reduced ring. is an -compatible ring, hence is -rigid, by [14, Lemma 2.2]. Let , such that . Then or , by a same way as used in [3, Proposition 6]. Thus is domain and is a completely prime ideal of .

Corollary 2.14. Let be an automorphism on . If is an -rigid ideal of , then .

Proof. is -rigid, hence is a completely semiprime -compatible ideal of , by Proposition 2.3. Therefore , by Theorem 2.13.

Theorem 2.15. Let be an automorphism of . If is a strongly (semi)prime -compatible ideal of , then is a strongly (semi)prime ideal of .

Proof. By Corollary 2.7, is a prime ideal of . Hence is a prime ring. We claim that zero is the only nil ideal of . Let be a nil ideal of . Assume be the set of all leading coefficients of elements of . First we show that is an ideal of . Clearly, is a left ideal of . Let and . Then there exists . Hence , for some nonnegative integers . Thus , since it is the leading coefficient of . Therefore , since is -compatible. Consequently, is an ideal of . Clearly is a nil ideal of . Hence and so . Therefore is a strongly prime ideal of .

Theorem 2.16. Let be an automorphism of . If each minimal strongly prime ideal of is -compatible, then .

Corollary 2.17. Let be an automorphism of . If is an -rigid ideal of , then .

Proof. is -rigid, hence is a completely semiprime -compatible ideal of , by Proposition 2.3, and that is a strongly semiprime ideal of . Therefore , by Theorem 2.15.

Example 2.12 also shows that there exists a ring such that all minimal strongly prime ideals are -compatible, but are not -rigid.

Theorem 2.18. Assume each minimal prime ideal of is -compatible. Then the following areequivalent:
(1) is completely semiprime;(2) and is completely semiprime.

Proof. (1)(2). Suppose that is a completely semiprime ideal of . It is enough to show that , by Theorem 2.8. Let be a minimal prime ideal of and . Since is a completely semiprime ideal of , is a completely semiprime ideal of . Clearly is an -invariant ideal of . Hence . We claim that is a minimal prime ideal of . Since is completely prime, is a completely prime ideal of . Let be a minimal prime ideal of such that . By assumption, is -compatible. Hence is a prime -compatible ideal of . Thus is a prime ideal of and . Since is a minimal prime ideal of , hence . Therefore and that . Consequently and that . Since and is completely semiprime, hence is completely semiprime.
(2)(1). Since is -compatible and completely semiprime, is an -compatible completely semiprime ideal of . Hence is a completely semiprime ideal of . Thus is a completely semiprime ideal of .

Corollary 2.19. Let be an automorphism of . Let each minimal prime ideal of be -compatible. Then the following are equivalent:
(1) is completely semiprime;(2) and is completely semiprime.

Acknowledgments

The author thanks the referee for his valuable comments and suggestions. This research is supported by the Shahrood University of Technology of Iran.

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