Uniformly Primal Submodule over Noncommutative Ring

<jats:p>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 <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1">
                        <mi>B</mi>
                     </math>
                  </jats:inline-formula> of <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2">
                        <mi>R</mi>
                     </math>
                  </jats:inline-formula> is uniformly not right prime to <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3">
                        <mi>N</mi>
                     </math>
                  </jats:inline-formula>, if there exists an element <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M4">
                        <mi>s</mi>
                        <mo>∈</mo>
                        <mi>M</mi>
                        <mo>−</mo>
                        <mi>N</mi>
                     </math>
                  </jats:inline-formula> with <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M5">
                        <mtext>sRB</mtext>
                        <mo>⊆</mo>
                        <mi>N</mi>
                     </math>
                  </jats:inline-formula>.The set <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M6">
                        <mtext>adj</mtext>
                        <mfenced open="(" close=")" separators="|">
                           <mrow>
                              <mi>N</mi>
                           </mrow>
                        </mfenced>
                        <mo>=</mo>
                        <mfenced open="{" close="}" separators="|">
                           <mrow>
                              <mi>r</mi>
                              <mo>∈</mo>
                              <mi>R</mi>
                           </mrow>
                           <mi>|</mi>
                           <mtext>mRr</mtext>
                           <mo>⊆</mo>
                           <mi>N</mi>
                           <mtext> </mtext>
                           <mtext>for some </mtext>
                           <mi>m</mi>
                           <mo>∈</mo>
                           <mi>M</mi>
                        </mfenced>
                     </math>
                  </jats:inline-formula> is uniformly not prime to <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M7">
                        <mi>N</mi>
                     </math>
                  </jats:inline-formula>.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.</jats:p>


Introduction
roughout 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]. e concept of uniformly primal ideal has been introduced and studied by Barnes [2]. Let A be an ideal of R. e ideal B of R is uniformly not right prime to A, if there exists an element y ∈ R − A with yRB ⊆ A. A is called uniformly primal if adj(A) is uniformly not right prime to A where adj(A) � x ∈ R|yRx ⊆ A for some y ∈ R − A . e prime avoidance theorem for rings with identity [3] states that if an ideal I of a ring is contained in a union of a finite number of prime ideals (P 1 , P 2 , . . . , P n ), then I must be contained in P k for some k ∈ 1, 2, . . . , n { }. Karamzadeh [4] generalizes the prime avoidance theorem for any ring that is not necessarily commutative. e 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. e 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 adj(N) � x ∈ R|mRr ⊆ N for some m ∈ M { } is uniformly not prime to N, where the subset B of R is uniformly not right prime to N if there exists an element s ∈ M − N with sRB ⊆ N. In particular, a number of papers concerning primal submodules have been studied by various authors (see, for example, [6][7][8][9][10]). In Section 2, we give some basic results about uniformly primal submodules and show that N 1 , N 2 , . . . , N n is a finite collection of uniformly primal submodules of an R-module M with adj(N j ) � P j for every j and Also, we study the prime avoidance theorem for modules over noncommutative rings and generalize it to the uniformly primal avoidance theorem for modules.

Uniformly Primal Ideal
e concept of primal ideals over noncommutative has been introduced and studied by Fuchs [11]. Definition 1. Let A be an ideal of R. e adjoint of A is the set of all elements of R that are not right prime to A and denoted by adj(A). In other words, adj(A) � x ∈ R|yRx ⊆ A for some y ∈ R − A}.
e ideal A of R is said to be primal if adj(A) forms an ideal of R. In this case, the adjoint of A will also be called the adjoint ideal of A. Definition 3. e ideal B of R is uniformly not right prime to A if there exists an element y ∈ R − A with yRB ⊆ A. Definition 4. An ideal A of a ring R is said to be uniformly primal if adj(A) is uniformly not right prime to A.
Proposition 1 (see [2]). If A is a uniformly primal ideal in R, then adj(A) is a prime ideal of R.
Proposition 2 (see [3]). If P, P 1 , and P 2 are ideals of R such that P⊆P 1 ∪ P 2 , then either P⊆P 1 or P⊆P 2 . e 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 P, P 1 , P 2 , . . . , P s be ideals of a ring R. e covering P ⊆P 1 ∪ P 2 ∪ · · · ∪ P s of P is called efficient if P is not contained in the union of any s − 1 of the ideals P i ′ s. Analogously, we shall say P � P 1 ∪ P 2 ∪ · · · ∪ P s is an efficient union if none of the P i ′ s 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. e following very important lemma is based on McCoy over commutative rings (see [3]).
Proof. Since P⊆P 1 ∪ P 2 ∪ · · · ∪ P s is an efficient covering, P⊆(P ∩ P 1 ) ∪ (P ∩ P 2 ) ∪ · · · ∪ (P ∩ P s ) is an efficient union. Now, by Lemma 1, P ∩ ∩ j≠k P j � ∩ j≠k (P ∩ P j )⊆P k . To prove the uniformly primal avoidance theorem for rings, we need the following result on the uniformly primal ideal. □ Proof. Suppose that some P k is uniformly primal ideal. SinceP⊆P 1 ∪ P 2 ∪ · · · ∪ P s is an efficient covering, there exists an element a k ∈ P − P k . If i ≠ j, then adj(P i )⊄adj(P k ), so there exists r j ∈ adj(P j ) such that r j ∉ adj(P k ). Since P k is a uniformly primal ideal, then by Proposition 1, adj(P k ) is a prime ideal of R. erefore, r � i≠k r i ∈ adj(P j ), but r ∉ adj(P k ). Consequently, e k Rr⊆P ∩ P j for every k ≠ j, but e k Rr⊄P k , which contradicts the fact that P ∩ ∩ j≠k P j � ∩ j≠k (P ∩ P j )⊆P k (by Corollary 1). erefore, no P k 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 P 1 , P 2 , . . . , P s be a finite number of ideals of a ring R and P be an ideal of R such that P⊆P 1 ∪ P 2 ∪ · · · ∪ P s . Assume that at least two of the P i ′ s are not uniformly primal and that adj(P i )⊄adj(P j ) whenever i ≠ j.
Proof. For the given covering P⊆P 1 ∪ P 2 ∪ · · · ∪ P s , let P⊆P α 1 ∪ P α 2 ∪ · · · ∪ P α m be its efficient reduction. en, 1 ≤ m ≤ s and m ≠ 2 where if m � 2, then by Proposition 2, P⊆P 1 or P⊆P 2 . If m > 2, then there exists at least one P α r to be uniformly primal ideal. By Proposition 3, this is impossible

Uniformly Primal Submodule
e concept of primal submodules has been introduced and studied by Dauns in [5].   Proof. Let a ∈ f(adj (N)). Since f is a module isomorphism, then f − 1 (a) ∈ adj(N) so that there exist s ∈ M − Nsuch that sRf − 1 (a)⊆N. Since f is a module isomorphism, then ∈ f(adj(N)). us, f(adj(N))⊆adj(f(N)). □ Proof. Let a, b ∈ (adj(f(N))). Since f is a module isomorphism, then by Proposition 6, we have f − 1 (a), Proof. Assume L⊄N; then, there is l ∈ L − N. For each a ∈ I, lRa⊆LI⊆Nwhile l ∉ N; thus, a ∈ adj(N). Callialp and Takir introduced the following definition (see [13]). □ Definition 11. Let N, N 1 , N 2 , . . . , N n be submodules of an R-module M. e covering N⊆N 1 ∪ N 2 ∪ · · · ∪ N n of N is called efficient if N is not contained in the union of any n − 1 of the submodulesN i ′ s. Analogously, we shall say N⊆N 1 ∪ N 2 ∪ · · · ∪ N n is an efficient union if none of the N i ′ s may be excluded. □ Proposition 10 (see [13]). Let N⊆N 1 ∪ N 2 ∪ · · · ∪ N n be an efficient cover of submodules of an R-module M where n > 2.
Proof. Let j ∈ 1, 2, . . . , n { }. Put ∩ i≠j (N i : M) � P j . By Proposition 9, ( ∩ i≠j N i : M) � P j . So, MP j ⊆ ∩ i≠j N i , and thus NP j ⊆ ∩ i≠j N i . But NP j ⊆N; then, NP j ⊆N ∩ ( ∩ i≠j N i )⊆N j by Proposition 10. is implies either N⊆N j or P j ⊆adj(N j ). But N⊄N j , and this implies that ∩ i≠j (N i : M)⊆adj(N j ). N 1 , N 2 , ..., N n are submodules of M such that N⊆N 1 ∪ N 2 ∪ · · · ∪ N n and ∩ i≠j (N i : M)⊄adj(N j ) for all j � 1, 2, . . . , n except possibly for at most two of the j's, then N⊆N k for some k ∈ 1, 2, . . . , n { }.

Theorem 2. Let N be a submodule of an R-module M. If
Proof. For the given covering N⊆N 1 ∪ N 2 ∪ · · · ∪ N n , let N⊆N α 1 ∪ N α 2 ∪ · · · ∪ N α m be its efficient reduction. en, 1 ≤ m ≤ n and m ≠ 2. If m > 2, then there exists at least one N i j satisfying ∩ i≠j (N α i : M)⊆adj(N α j ) which is contradiction to Lemma 2. Hence, m � 1, so N⊆N α 1 � N k for some k ∈ 1, 2, . . . , n { }. Bland in [14] proved the following result.  Corollary 4 (see [5]). Let M be an R-module. If N is a prime submodule of M, then N 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]). en, 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 a, b ∈ S, there exists r ∈ R such that arb ∈ S.

Journal of Mathematics
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 a ∈ S and m ∈ N, there exists r ∈ R such that mra ∈ N.
Proposition 11. Let M be an R-module and N be a uniformly primal submodule of M. en, M − N is an S-system where S � R − adj(N).
Proof. Since N is a uniformly primal submodule of M, by Proposition 5, adj(N) is a prime ideal of R. Let a ∈ S � R − adj(N) and t ∈ M − N, so tRa⊄N. erefore, M − N is an S-system. □ Proposition 12. Let P 1 , P 2 , . . . , P n be a finite number of prime ideals in a ring R and S � R − ∪ n i�1 P i . en, S is an m-system subset of R.
Proof. Let a, b ∈ S and assume on the contrary that aRb⊄S; thus, aRb ⊆ ∪ n i�1 P i . en, RaRbR ⊆ ∪ n i�1 P i . Hence, by the prime avoidance theorem for rings (see [3] and [4]), we have (RaR)(RbR) � RaRbR ⊆ P i for some i � 1, 2, . . . , n. Since P i is prime, then either RaR ⊆ P i or RbR ⊆ P i . If RaR ⊆ P i , then a ∉ S which is a contradiction. Similarly, if RbR ⊆ P i , then there exists r ∈ R such that arb ∈ S. erefore, S is an m-system subset of R. □ Proposition 13. Let N 1 , N 2 , . . . , N n be a finite collection of uniformly primal submodules of an R-module M with adj(N j ) � P j for every j and (N j : M)⊄adj(N k ) whenever j ≠ k. en, S * � M − ∪ n i�1 N i is an S-system subset of M, where S � R − ∪ n i�1 P i .
Proof. By Proposition 12, S is an m-system subset of R, so to prove S * is an S-system, let a ∈ S and m ∈ S * and assume on the contrary that mRa⊄S * ; thus, mRa⊆ ∪ n i�1 N i . en, mRaR⊆ ∪ n i�1 N i , so by eorem 3 (uniformly primal avoidance theorem for modules), we have mRaR⊆N i and mRa⊆N i for some i � 1, 2, . . . , n. Since N i is uniformly primal, then m ∈ N i or a ∈ adj(N i ). If m ∈ N i , then m ∉ S * which is a contradiction. If a ∈ adj(N i ), then a ∉ S. en, there exists r ∈ R such that mra ∈ S * . erefore, S * is an S-system subset of M.

Data Availability
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Conflicts of Interest
e author declares that there are no conflicts of interest.