On the primeness of near-rings Primeness of near-rings

In this paper, we study the different kinds of the primeness on the class of near-rings and we give new characterizations for them. For that purpose, we introduce new concepts called set-divisors, ideal-divisors, etc. and we give equivalent statements for 3-primeness which make 3-primeness looks like the forms of the other kinds of primeness. Also, we introduce a new different kind of primeness in near-rings called K-primeness which lies between 3-primeness and e-primeness. After that, we study different kinds of prime ideals in near- rings and find a connection between them and new concepts called set-attractors, ideal-attractors, etc. to make new characterizations for them. Also, we introduce a new different kind of prime ideals in near-rings called K-prime ideals.


Introduction
We say that R is a right (left) near-ring if ðR; þÞ is a group, ðR; $Þ is a semigroup and R satisfies the right (left) distributive law. Throughout this paper, R will be a left near-ring. We say that R is an abelian near-ring if x þ y ¼ y þ x for all x; y ∈ R and we say that R is a commutative near-ring if xy ¼ yx for all x; y ∈ R. A zero-symmetric element is an element x ∈ R satisfying 0x ¼ 0. A near-ring R is called a zero-symmetric near-ring, if 0x ¼ 0 for all x ∈ R. A constant element is an element y ∈ R satisfying zy ¼ y for all z ∈ R. An element x ∈ R is called a right (left) zero divisor in R if there exists a non-zero element y ∈ R such that yx ¼ 0 (xy ¼ 0). A zero divisor is either a right or a left zero divisor. By a near-ring without zero divisors, we mean a near-ring without non-zero divisors of zero. If A and B are two non-empty subsets of R, then the product AB means the set fabja ∈ A; b ∈ Bg. We say that U is a right (left) R-subgroup of R, if U is a subgroup of ðR; þÞ satisfies UR ⊆ U (RU ⊆ U). We say that U is a two-sided R-subgroup of R, if U is both a right and a left R-subgroup of R. We say that I is a right (left) ideal of R, if I is a normal subgroup of ðR; þÞ satisfies ðr þ iÞs − rs ∈ I for all i ∈ I ; r; s ∈ R (RI ⊆ I). We say that I is an ideal of R if it is both a right and a left ideal of R. We say that U is a semigroup right (left) ideal of R, if U is a non-empty subset of R satisfies UR ⊆ U (RU ⊆ U). We say that U is a semigroup ideal of R if it is both a semigroup right and left ideal of R (some authors call U a right (left, two-sided) R-subset of R [8]). For any group ðG; þÞ, M ðGÞ denotes the near-ring of all maps from G to G with the two operations of addition and composition of maps. M o ðGÞ is the zero-symmetric subnear-ring of M ðGÞ consisting of all zero preserving maps from G to itself (and to make them left near-rings we should write f ðgÞ by gf, where f ∈ M ðGÞ or M o ðGÞ and g ∈ G ). A trivial zero-symmetric nearring R is a zero-symmetric near-ring such that the multiplication on the group ðR; þÞ is defined by xy ¼ y and 0y ¼ 0 for all x ∈ R − f0g; y ∈ R. A near-field N is a near-ring in which ðN − f0g; $Þ is a group. For further information about near-rings, see [8] and [9].
In near-rings, there are five well-known kinds of primeness. We say that: R is 0-prime (the usual primeness) if, for every two ideals I and J of R, IJ ¼ f0g if, for any 0 ≠ a; x; y ∈ R, xca ¼ yca for all c ∈ R implies x ¼ y. These five kinds of primeness are equivalent in the class of rings. But in the class of near-rings, we have: (1) R is equiprime implies that R is zero-symmetric 3-prime, (2) R is 3-prime implies that R is 2-prime, (3) R is zero-symmetric 2-prime implies that R is 1-prime and (4) R is 1-prime implies that R is 0-prime. For details about these kinds and their examples and relationships see [1][2][3][5][6][7] and [10]. A near-ring (a ring) R is called 3-semiprime (semiprime) if, for all x ∈ R, xRx ¼ f0g implies x ¼ 0. An ideal P of R is: (i) a 0-prime ideal of R if for every two ideals A and B of R, AB ⊆ P implies that A ⊆ P or B ⊆ P, (ii) a 1-prime ideal of R if for every two right ideals A and B of R, AB ⊆ P implies that A ⊆ P or B ⊆ P, (iii) a 2-prime ideal of R if for every two right R-subgroups A and B of R, AB ⊆ P implies that A ⊆ P or B ⊆ P, (iv) a 3-prime ideal of R if for a; b ∈ R, aRb ⊆ P implies that a ∈ P or b ∈ P, (v) an e-prime (equiprime) ideal of R if for every a ∈ R − P and x; y ∈ R, xca − yca ∈ P for all c ∈ R implies that x − y ∈ P. Clearly that any nearring is a υ-prime ideal of itself, where υ ∈ f0; 1; 2; 3; eg. It is well-known that (ii) implies (i) and (iv) implies (iii). Also, for zero-symmetric near-rings we have (iii) implies (ii). An ideal I of R is called completely prime if, for a; b ∈ R, ab ∈ I implies that a ∈ I or b ∈ I. If the zero ideal is completely prime, then we say that R is completely prime. Then R is completely prime if and only if R is without zero divisors. For more details about prime ideals, see [2,4,5] and [10].
In [1], the authors gave us a short historical view about the primeness of near-rings. We will use it and add some information to it.
In this paper we extend the idea of primeness that they did and give some new results for the primeness of near-rings. Firstly, we introduce new concepts called set-divisors, ideal divisors, etc. These concepts are generalizations of the concept of zero divisors and give another characterization of different kinds of the primeness in near-rings and hence in rings. Also, we study the 3-primeness and give new characterizations of 3-prime (3-semiprime) nearrings and hence for prime (semiprime) rings. These characterizations make 3-primeness looks like the forms of the other kinds of primeness. In fact, we show that a near-ring (a ring) is 3-prime (prime) if and only if UV ¼ f0g implies U ¼ f0g or V ¼ f0g, where U and V are semigroup left ideals of R. Hence, a ring is prime if and only if it is without zero-semigroup right (left) ideal divisors. A similar result is made for 3-semiprime near-rings (semiprime rings) and we conclude that: for a near-ring R, if r 2 ≠ 0 for all r ∈ R − f0g, then R is 3-semiprime. We show that some kinds of near-rings are 3-prime if and only if they are 2-prime. Also, we introduce a new kind of primeness in near-rings (the sixth one) called K-primeness and we show that it is totally different from the other kinds of primeness and it lies between 3-primeness and e-primeness. Depending on that, we give two chains of primeness in the class of zero-symmetric near-rings for comparison. In the last part of the paper, we study different kinds of prime ideals. We introduce a new kind of prime ideals called K-prime ideals and we show that they are different from the other kinds of prime ideals. they lie between 3-prime ideal and e-prime ideals. Also, we give a new characterization of 3-prime ideals and show that P is a 3-prime ideal of R if and only if UV ⊆ P implies U ⊆ P or V ⊆ P, where U and V are semigroup left ideals of R. We introduce new concepts called set-attractors, ideal-attractors, etc. which are generalizations of the new concepts above (set-divisors, etc.). We make a connection between these concepts and different kinds of prime ideals in near-rings to give a new characterization of these prime ideals. Finally, we use these concepts to show that: P is a completely prime ideal of R if and only if R is without external P set-attractors. (1) Let A be a non-empty subset of R. We say that A is a left zero-set divisor (a right zeroset divisor) of R if there exists a non-empty non-zero subset B of R such that AB ¼ f0g

On prime near-rings
(2) Let A be an ideal of R. We say that A is a left zero-ideal divisor (a right zero-ideal divisor) of R if there exists a non-zero ideal B of R such that AB ¼ f0g (BA ¼ f0g). We say that A is a zero-ideal divisor of R if A is a left or a right zero-ideal divisor of R.
We can do same definitions if A is a left (right) ideal, a left (right) R -subgroup, a two-sided R-subgroup, a semigroup left (right) ideal or a semigroup ideal. Definition 2.1 generalizes the concept of zero divisors in rings and near-rings. So, we have the following remark.
Remark 2.1. From Definition 2.1, we can rewrite the definitions of different kinds of the primeness as follows: Let R be a near-ring. Then (1) R is completely prime if and only if R is without zero divisors if and only if R is without zero-set divisors.
Remark 2.1 enhances a question: Can we get a definition of 3-primeness like that mentioned in Remark 2.1? The following result answers this question.
Theorem 2.1. Let R be a near-ring. Then the following statements are equivalent:
To prove that (iv) implies (i), we will use the contradiction. For that purpose, suppose R is not 3-prime. So there exist non-zero elements x; y ∈ R such that xRy ¼ f0g. Thus, RxRy ¼ f0g. But Rx and Ry are semigroup left ideals of R, so Rx ¼ f0g or Ry ¼ f0g by (iv). Hence, Rf0; xg ¼ f0g or Rf0; yg ¼ f0g and either f0; xg or f0; yg is a semigroup left ideal of R. But R is also a semigroup left ideal of R. Thus, f0; xg ¼ 0, f0; yg ¼ f0g or R ¼ f0g by (iv), a contradiction with that x; y; R are all non-zero. So R is 3-prime and (iv) implies (i). -For zero-symmetric near-rings, we have the following extra result.
Theorem 2.2. Let R be a zero-symmetric near-ring. Then the following statements are equivalent: Now, we can add (5)  Since any ring is a zero-symmetric near-ring, we have the following result: A ring is prime if and only if it is without zero-semigroup right (left) ideal divisors.
Using the same idea, the following result gives us a result for 3-semiprime zero-symmetric near-rings.
Theorem 2.4. Let R be a zero-symmetric near-ring. Then the following statements are equivalent: Proof. (i) implies (ii). Suppose (i) holds. Let U be a semigroup left ideal of R such that aU ¼ f0g, where a ∈ U. Then for all v ∈ U, we have aRv ¼ f0g. Thus, aRa ¼ f0g and a ¼ 0 by (i). (i) implies (iii) can be proved by the same way. (ii) implies (iv) and (iii) implies (v) are clear. (iv) implies (v). Suppose that (iv) holds and U 2 ¼ f0g, where U is a semigroup right ideal of R. So uRu ¼ f0g for all u ∈ U and hence RuRu ¼ f0g. But Ru is a semigroup left ideal of R. So Ru ¼ f0g for all u ∈ U by (iv). So f0; ug is a semigroup left ideal of R and f0; ugf0; ug ¼ f0g for all u ∈ U. So u ¼ 0 by (iv) and hence U ¼ f0g.

Corollary 2.5. A ring R is semiprime if and only if U
But in the general case of 3-semiprime near-rings, we have only the following result.
Theorem 2.6. Let R be a near-ring. Then the following statements are equivalent: Unfortunately, we cannot remove the word "zero-symmetric" in Theorems 2.2 and 2.4. The following example is the near-ring in [9,Appendix,E,22] and it shows that the condition "zero-symmetric" in Theorems 2.2 and 2.4 is not redundant. Example 1. Let ðR; þÞ be the Klein's four group f0; a; b; cg. Then it is an abelian group such that x þ x ¼ 0 for all x ∈ R and x þ y ¼ z for all different non-zero elements x; y; z ∈ R. Define the multiplication on R as follows: Clearly R is an abelian non-zero-symmetric near-ring. The only semigroup right ideals of R are R, f0; ag and f0; a; bg. So R satisfies the conditions "UV ¼ f0g implies U ¼ f0g or V ¼ f0g, where U and V are semigroup right ideals of R" and "U 2 ¼ f0g implies U ¼ f0g, where U is a semigroup right ideal of R". But R is not 3-semiprime as bRb ¼ f0g. From Theorem 2.6, we can deduce that there is a non-zero semigroup left ideal V of R such that V 2 ¼ f0g and It is easy to find out that V ¼ f0; bg and v ¼ b.
From the above example, observe that f0; a; bgb ¼ f0; a; bgfbg ¼ f0g: So, we cannot use this example for (ii) or (iii) in Theorem 2.2 and for (iii) in Theorem 2.4. In fact, removing "zero-symmetric" from those parts is an open problem.
Corollary 2.7. Let R be a near-ring. If r 2 ≠ 0 for all r ∈ R − f0g, then R is 3-semiprime.
Proof. Suppose there exists a non-zero semigroup left ideal U of R such that aU ¼ f0g, where a ∈ U. That means a 2 ¼ 0. By hypothesis, a ¼ 0 and hence R is 3-semiprime. - Then R is semiprime since r 2 ≠ 0 for all r ∈ R − f0g.
Then R is a prime ring and hence semiprime, but For commutative near-rings, we have the converse and we get the following result.
Corollary 2.8. Let R be a commutative near-ring. Then r 2 ≠ 0 for all r ∈ R − f0g if and only if R is 3-semiprime. We conclude this section by the following results about the relation between 2-primeness and 3-primeness. The fact that R is 3-prime implies R is 2-prime is well-known. The following results have the converse. Theorem 2.9. Let R be a zero-symmetric near-ring such that 2R ¼ f0g. Then R is 3-prime if and only if R is 2-prime.
Proof. Suppose that xRy ¼ f0g. Thus, xRyR ¼ f0g. But xR and yR are right R-subgroups of R. So xR ¼ f0g or yR ¼ f0g as R is 2-prime. Hence, f0; xgR ¼ f0g or f0; ygR ¼ f0g and then either f0; xg or f0; yg is a right R-subgroup of R. But R is also a right R-subgroup of R. Thus,

Theorem 2.10 Any distributive near-ring R is 3-prime if and only if it is 2-prime.
Proof. Suppose that R is 2-prime and xRy ¼ f0g for some x; y ∈ R. So xRyR ¼ f0g and hence xR ¼ f0g or yR ¼ f0g. So AR ¼ f0g or BR ¼ f0g, where A ¼ fnxjn ∈ ℤg and B ¼ fnyjn ∈ ℤg. So A and B are right R-subgroups of R and hence A ¼ f0g or B ¼ f0g. Therefore, x ¼ 0 or y ¼ 0 and R is 3-prime. -

K-prime near-rings
In this section, we will introduce a new kind of primeness of near-rings called K-primeness. Firstly, we will begin with the following result. (ii) for any 0 ≠ a; x; y ∈ R, xsa ¼ yra for all s; r ∈ R − f0g implies x ¼ y.

Proof.
A ring R is prime if and only if it is equiprime, so we will use the definition of equiprimeness, i.e. for any 0 ≠ a; x; y ∈ R, xca ¼ yca for all c ∈ R implies x ¼ y.
Definition 3.1. Let R be a near-ring. We say that R is K-prime if, for any 0 ≠ a; x; y ∈ R, xsa ¼ yra for all s; r ∈ R − f0g implies x ¼ y.
As we mentioned before for rings, a ring is prime if and only if it is equiprime. So we have the following result.

Corollary 3.2. A ring R is prime if and only if it is K-prime.
The following result shows that every K-prime near-ring is zero-symmetric 3-prime.
Theorem 3.3. Let R be a K-prime near-ring. Then R is zero-symmetric 3-prime.
Proof. Firstly, we will show that R is zero-symmetric. If R is not zero-symmetric, then it has at least one non-zero constant element c (see [8,Theorem 1.15). For different elements x; y of R, we have that xsc ¼ yrc ¼ c for all s; r ∈ R − f0g, a contradiction with the hypothesis. So R is zero-symmetric. Now, suppose xRy ¼ f0g for some x; y ∈ R. So xcy ¼ 0 for all c ∈ R. If y ≠ 0, then xcy ¼ 0ry for all c; r ∈ R. So x ¼ 0 from the hypothesis and hence R is 3-prime. -In the case of near-rings, we have only that e-primeness implies K-primeness as shown in the proof of Theorem 3.1 (since an e-prime near-ring is zero-symmetric [10]). But the converse is not true as we will show in the next example. We will use the near-ring mentioned in [9,Appendix,F,7] in the next example. So R is an abelian near-ring which is not a ring (as Clearly that is true if x or y is equal to zero, since R is without zero divisors. That is the only possible case. In fact, if xsa ¼ yra for all s; r ∈ R − f0g and x; y; a are all non-zero, then from the table we can choose s o ; r o ∈ R − f0g to satisfy that xs o ¼ 1 and yr o ¼ 2. Hence, a ¼ 2a which implies that a ¼ 0 (from the table), a contradiction with 0 ≠ a. Therefore, K-primeness does not imply e-primeness. Also, we can find zero-symmetric 3-prime near-rings which are not K-prime, as the following example shows.
Example 6. Let R be a trivial zero-symmetric near-ring of order greater than 2. Clearly R is 3prime. Taking two non-zero elements x and y such that x ≠ y, we have xsx ¼ yrx ¼ x for all s; r ∈ R − f0g. So R is not K-prime. Theorem 3.1,Theorem 3.3 and the examples after them show that K-primeness is a new kind of primeness.
Observe that K-primeness lies between 3-primeness and e-primeness (equiprimeness). So we have the following chain of primeness in the class of zero-symmetric near-rings: The class of e-prime near-rings ⊆ The class of K-prime near-rings ⊆ The class of 3-prime near-rings ⊆ The class of 2-prime near-rings ⊆ The class of 1-prime near-rings ⊆ The class of 0-prime near-rings Remark 3.1. Observe that: (i) It is well-known that M o ðGÞ is e-prime (see [10]) and hence K-prime. Observe that it has zero divisors. (ii) Since M ðGÞ is not zero-symmetric, so it is not K-prime (and hence not e-prime), but it has zero divisors.
Primeness of near-rings (iii) Let N be any near-field. Then N is e-prime and hence K-prime. Indeed, for any 0 ≠ a; x; y ∈ R such that xca ¼ yca for all c ∈ R, we have that x ¼ y by choosing c ¼ a −1 . Observe that N is without zero divisors.
(iv) Example 6 shows a 3-prime near-ring without zero divisors which is not K-prime (and hence not e-prime).
From the above parts in Remark 3.1, there is no relation between e-primeness (Kprimeness) and the existence of zero divisors in near-rings. So, we have another chain of the primeness in the class of zero-symmetric near-rings: The class of completely prime near-rings ⊆ The class of 3-prime near-rings ⊆ The class of 2-prime near-rings ⊆ The class of 1-prime near-rings ⊆ The class of 0-prime near-rings

On prime ideals
The next definition introduces K-prime ideals.
Definition 4.1. Let R be a near-ring and P an ideal of R. Then P is a K-prime ideal of R if for every a ∈ R − P and x; y ∈ R, xra − ysa ∈ P for all r; s ∈ R − P implies x − y ∈ P.
Clearly R is K-prime if and only if f0g is a K-prime ideal of R.
The relationship between K-prime ideals and other kinds of prime ideals is stated in the following result.
Theorem 4.1. Let R be a near-ring with an ideal P.
(i) If P is a K-prime ideal of R, then P is a 3-prime ideal of R.
(ii) If P is an e-prime ideal of R, then P is a K-prime ideal of R.
Proof. (i) Firstly, we will show that P contains all the constant elements of R. Let c be a constant element in R. If c ∈ R − P, then xrc À ysc ¼ c À c ¼ 0 ∈ P for all x; y ∈ R and r; s ∈ R − P. So x − y ∈ P and hence x − 0 ¼ x ∈ P for all x ∈ R. Thus, P ¼ R, a contradiction with c ∉ P. So c ∈ P. Now, suppose aRb ⊆ P for some a; b ∈ R and b ∉ P. From above, any element s ∈ R − P is a zero-symmetric element. So 0sb ¼ 0 ∈ P for all s ∈ R − P. So arb − 0sb ∈ P for all r; s ∈ R − P. Thus, a ∈ P by the hypothesis and P is 3-prime. (ii) Firstly, observe that if r ∈ P and s ∈ R is a zero-symmetric element, then rs ¼ ðr þ 0Þs À 0s ∈ P: Suppose xra − ysa ∈ P for all r; s ∈ R − P, where a ∈ R − P and x; y ∈ R. So xca − yca ∈ P for all c ∈ R − P. Now, suppose c ∈ P. As a ∉ P, we have that a is a zero-symmetric element (see [10]). So ca ∈ P and hence xca − yca ∈ P. But P is e-prime. So x − y ∈ P and P is a K-prime ideal of R. - The next result generalizes Theorem 2.1 for 3-prime ideals.
Theorem 4.2. Let R be a near-ring and P an ideal of R. Then the following statements are equivalent: (i) P is a 3-prime ideal of R. (ii) BU ⊆ P implies B ⊆ P or U ⊆ P, where B is a non-empty subset of R and U is a semigroup left ideal of R.
(iii) UV ⊆ P implies U ⊆ P or V ⊆ P, where U and V are semigroup left ideals of R.
if A ⊆ P. If A ? P, then we say that A is an external P set-attractor (P ideal-attractor, etc.) of R.
If R does not have any external P set-attractors (P ideal-attractors, etc.), then we say that R is without external P set-attractors (P ideal-attractors, etc.), i.e. for a P set-attractor (P idealattractor, etc.) A of R, we have that A ⊆ P Example 9. (i) Any near-ring R is without external (or internal) R-set attractors.
(ii) Any near-ring without zero divisors is without external f0g-set attractors. (iii) Let R be the ring ℤ 4 . Take P to be the ideal f0; 2g. Then R is without external P set-attractors.
(iv) Let R be the ring ℤ 6 . Take P to be the ideal f0g. Then f2g, f3g and f4g are external P set-attractors and f0g is an internal P set-attractor.
Theorem 4.4. Let R be a near-ring with an ideal P. Then the following statements are equivalent: (i) R is without external P set-attractors.
(ii) P is a completely prime ideal of R.
Proof. (i) implies (ii), Suppose (i) holds and ab ∈ P for some a; b ∈ R. So fagfbg ⊆ P. If a ∉ P, then b ∈ P by (i) and P is completely prime. (ii) implies (i). Suppose (ii) holds and A is a P set-attractor of R. So there exists a non-empty subset B of R and B ? P such that AB ⊆ P or BA ⊆ P. Suppose the case is AB ⊆ P. Take y ∈ B − P. So xy ∈ P for all x ∈ A and then A ⊆ P by (ii). By the same way we can do for the other case. So R is without external P set-attractors. - Let R be a near-ring with an ideal P. Then (1) P is completely prime if and only if R is without external P set-attractors if and only if for every two non-empty subsets A and B of R, AB ⊆ P implies A ⊆ P or B ⊆ P.
(2) P is 0-prime if and only if R is without external P ideal-attractors.
(3) R is 1-prime if and only if R is without external P right ideal-attractors.
(4) R is 2-prime if and only if R is without external P right R-subgroup-attractors.
(5) R is 3-prime if and only if R is without external P semigroup left ideal-attractors.