Uniformly Primal Submodule over Noncommutative Ring

Abstract

Let R be an associative ring with identity and M be a unitary right R-module. A submodule N of M is called a uniformly primal submodule provided that the subset of is uniformly not right prime to , if there exists an element with .The set is uniformly not prime to .This paper is concerned with the properties of uniformly primal submodules. Also, we generalize the prime avoidance theorem for modules over noncommutative rings to the uniformly primal avoidance theorem for modules.

1. Introduction

Throughout this paper, all rings are associative with identity and all modules are unitary modules. For detailed description regarding rings and modules, interested readers are encouraged to go through the book of Kelarev et al. [1]. The concept of uniformly primal ideal has been introduced and studied by Barnes [2]. Let A be an ideal of R. The ideal B of R is uniformly not right prime to A, if there exists an element with . A is called uniformly primal if is uniformly not right prime to A where . The prime avoidance theorem for rings with identity [3] states that if an ideal of a ring is contained in a union of a finite number of prime ideals , then must be contained in for some . Karamzadeh [4] generalizes the prime avoidance theorem for any ring that is not necessarily commutative. The aim of Section 1 is to generalize the prime avoidance theorem for rings over noncommutative rings to the uniformly primal avoidance theorem over noncommutative rings.

The concept of uniformly primal submodules has been introduced and studied by Dauns in [5]. A submodule N of M is called a uniformly primal submodule provided that the set is uniformly not prime to N, where the subset B of R is uniformly not right prime to N if there exists an element with . In particular, a number of papers concerning primal submodules have been studied by various authors (see, for example, [6–10]). In Section 2, we give some basic results about uniformly primal submodules and show that is a finite collection of uniformly primal submodules of an R-module M with for every j and whenever . Then, is an S-system subset of M, where . Also, we study the prime avoidance theorem for modules over noncommutative rings and generalize it to the uniformly primal avoidance theorem for modules.

2. Uniformly Primal Ideal

The concept of primal ideals over noncommutative has been introduced and studied by Fuchs [11].

Definition 1. Let A be an ideal of . The adjoint of is the set of all elements of that are not right prime to and denoted by . In other words, .

Definition 2. The ideal A of R is said to be primal if forms an ideal of R. In this case, the adjoint of A will also be called the adjoint ideal of A.

Definition 3. The ideal B of R is uniformly not right prime to A if there exists an element with .

Definition 4. An ideal A of a ring R is said to be uniformly primal if is uniformly not right prime to A.

Proposition 1. (see [2]). If A is a uniformly primal ideal in R, then is a prime ideal of R.

Proposition 2. (see [3]). If are ideals of R such that , then either or . The definition of efficient union of ideals was introduced in the rings that are commutative (see [12]). We give a generalization to it in rings that are not necessary commutative as follows.

Definition 5. Let be ideals of a ring R. The covering of P is called efficient if P is not contained in the union of any of the ideals. Analogously, we shall say is an efficient union if none of the may be excluded. Any cover or union consisting of ideals of R can be reduced to an efficient one, called an efficient reduction, by deleting any unnecessary terms. The following very important lemma is based on McCoy over commutative rings (see [3]).

Lemma 1. (see [3]). Let be an efficient union of ideals where s > 2. Then, for all . As an application, we obtain following corollary.

Corollary 1. Let be an efficient cover of ideals of a ring R where s > 2. Then, for all .

Proof. Since is an efficient covering, is an efficient union. Now, by Lemma 1, . To prove the uniformly primal avoidance theorem for rings, we need the following result on the uniformly primal ideal.

Proposition 3. Let be an efficient covering consisting of ideals where s > 2. If for every , then no for is a uniformly primal ideal of R.

Proof. Suppose that some is uniformly primal ideal. Since is an efficient covering, there exists an element . If , then , so there exists such that . Since is a uniformly primal ideal, then by Proposition 1, is a prime ideal of R. Therefore, , but . Consequently, for every , but , which contradicts the fact that (by Corollary 1). Therefore, no is a uniformly primal. Now, we will give the proof of the main theorem of this section.

Theorem 1. (uniformly primal avoidance theorem of rings). Let be a finite number of ideals of a ring R and P be an ideal of R such that . Assume that at least two of the are not uniformly primal and that whenever . Then, for some .

Proof. For the given covering , let be its efficient reduction. Then, and where if , then by Proposition 2, or . If , then there exists at least one to be uniformly primal ideal. By Proposition 3, this is impossible as if . Hence, , so for some k.

3. Uniformly Primal Submodule

The concept of primal submodules has been introduced and studied by Dauns in [5].

Definition 6. Let M be an R-module and N be a submodule of M. For any , the submodule is denoted by . Analogously, for a subset of ,  =  where .

Definition 7. Let M be an R-module and N be a submodule of M. The element is right prime to N if , i.e., if implies . The element is not right prime to N if , i.e., there exists an element with , since . A subset A of R is not right prime to N if for any , a is not right prime to N. In this case, we say that A is pointwise not right prime to N. The subset A of R is uniformly not right prime to N if there exists an element with , i.e., A is uniformly not right prime to N if and only if .

Definition 8. Let M be an R-module and N be a submodule of M. The adjoint of N is the set of all elements of R that are not right prime to N and denoted by . In the other words, .

Definition 9. Let M be an R-module. A proper submodule N of M is said to be primal if adj (N) forms an ideal of R. In this case, the adjoint of N will also be called the adjoint ideal of N.

Proposition 4. Let N be a submodule of an R-module M. If is uniformly not prime to N, then is an ideal of R, and as a consequence, N is a primal submodule.

Definition 10. Let M be an R-module. A proper submodule N of M is said to be uniformly primal if is uniformly not prime to N.

Proposition 5. (see [8]). Let be an -module. If is a uniformly primal submodule of , then is a prime ideal of . In the following propositions, we show the behavior of a primal submodule under isomorphism.

Proposition 6. Let be a module isomorphism. If N is a primal submodule of M, then .

Proof. Let . Since f is a module isomorphism, then so that there exist such that . Since f is a module isomorphism, then , but so that which implies that . Thus, . Now, let so that there exist such that . Since f is a module isomorphism, then , but so that which implies that . That is, . Thus, .

Proposition 7. Let be a module isomorphism. If is a primal submodule of , then is a primal submodule of .

Proof. Let . Since is a module isomorphism, then by Proposition 6, we have . But is a primal submodule; then, . Also, since is a module isomorphism, we have . Hence, by Proposition 6, where is a primal submodule of .

Proposition 8. Let and be proper submodules of an -module and be an ideal of . If , then either or .

Proof. Assume ; then, there is . For each , while ; thus, . Callialp and Takir introduced the following definition (see [13]).

Definition 11. Let be submodules of an -module . The covering of is called efficient if is not contained in the union of any of the submodules. Analogously, we shall say is an efficient union if none of the may be excluded.

Proposition 9. If and are submodules of an -module , then.

Proof. Let . Then, . Thus, and ; hence, . Therefore, . Now, if , then so that and ; thus, and then . Therefore, .

Proposition 10. (see [13]). Let be an efficient cover of submodules of an R-module M where . Then, for all . Now, by using Propositions 9 and 10 we will prove the following lemma and theorem.

Lemma 2. Let be an efficient cover of submodules of an -module where ; then, for all , .

Proof. Let . Put . By Proposition 9, . So, , and thus . But ; then, by Proposition 10. This implies either or . But , and this implies that.

Theorem 2. Let N be a submodule of an R-module M. If are submodules of M such that and for all except possibly for at most two of the j’s, then for some

Proof. For the given covering , let be its efficient reduction. Then, and . If , then there exists at least one satisfying which is contradiction to Lemma 2. Hence, , so for some
Bland in [14] proved the following result.

Corollary 2. Let P be a prime ideal of a ring R and suppose that A and B are ideals in R. If , then either . The following corollary follows immediately from Proposition 5 and Corollary 2.

Corollary 3. Let be uniformly primal submodules. Then, the following two conditions are equivalent:(a)(b)Now, the main theorem of this section is uniformly primal avoidance theorem which follows immediately from Proposition 5, Theorem 2, and Corollary 3.

Theorem 3. (uniformly primal avoidance theorem for modules). Let N be a submodule of an R-module M. are submodules of M such that . Assume that at most two of the are not uniformly primal and ; then, for some

Corollary 4. (see [5]). Let be an -module. If is a prime submodule of , then is primal. Since every prime submodule is uniformly primal by Corollary 4, then the uniformly primal avoidance theorem is a generalization of the prime avoidance theorem for modules. Now, we will recall the concept of S-system subsets of modules, which was introduced in [11] (also see [13,15]). Then, we will prove some results on the S-system and uniformly primal submodule.

Definition 12. A nonempty subset S of a ring R is said to be an m-system if for any , there exists such that .

Definition 13. Let M be an R-module and S be an m-system. A nonempty subset N of R-module M is said to be a S-system if for any and , there exists such that .

Proposition 11. Let M be an R-module and N be a uniformly primal submodule of M. Then, is an S-system where

Proof. Since N is a uniformly primal submodule of M, by Proposition 5, is a prime ideal of R. Let and , so . Therefore, is an S-system.

Proposition 12. Let be a finite number of prime ideals in a ring R and . Then, S is an m-system subset of R.

Proof. Let and assume on the contrary that ; thus, . Then, . Hence, by the prime avoidance theorem for rings (see [3] and [4]), we have for some . Since is prime, then either or . If , then which is a contradiction. Similarly, if , then there exists such that . Therefore, S is an m-system subset of R.

Proposition 13. Let be a finite collection of uniformly primal submodules of an R-module M with for every and . Then, is an S-system subset of M, where .

Proof. By Proposition 12, S is an m-system subset of R, so to prove is an S-system, let and and assume on the contrary that ; thus, . Then, , so by Theorem 3 (uniformly primal avoidance theorem for modules), we have and for some . Since is uniformly primal, then or . If , then which is a contradiction. If , then . Then, there exists such that . Therefore, is an S-system subset of M.

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Conflicts of Interest

The author declares that there are no conflicts of interest.