1. Introduction
We say that R is a right (left) near-ring if (R,+) is a group, (R,⋅) 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+y=y+x for all x,y∈R and we say that R is a commutative near-ring if xy=yx for all x,y∈R. A zero-symmetric element is an element x∈R satisfying 0x=0. A near-ring R is called a zero-symmetric near-ring, if 0x=0 for all x∈R. A constant element is an element y∈R satisfying zy=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=0 (xy=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 {ab|a∈A,b∈B}. We say that U is a right (left) R-subgroup of R, if U is a subgroup of (R,+) 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 (R,+) satisfies (r+i)s−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 (G,+), M(G) denotes the near-ring of all maps from G to G with the two operations of addition and composition of maps. Mo(G) is the zero-symmetric subnear-ring of M(G) consisting of all zero preserving maps from G to itself (and to make them left near-rings we should write f(g) by gf, where f∈M(G) or Mo(G) and g∈G). A trivial zero-symmetric near-ring R is a zero-symmetric near-ring such that the multiplication on the group (R,+) is defined by xy=y and 0y=0 for all x∈R−{0},y∈R. A near-field N is a near-ring in which (N−{0},⋅) 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={0} implies I={0} or J={0}, R is 1-prime if, for every two right ideals K and L of R, KL={0} implies K={0} or L={0}. R is 2-prime if, for every two right R-subgroups A and B of R, AB={0} implies A={0} or B={0}. R is 3-prime if, for all x,y∈R, xRy={0} implies x=0 or y=0 and R is equiprime (e-prime) if, for any 0≠a,x,y∈R, xca=yca for all c∈R implies x= 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 [1–3,5–7] and [10]. A near-ring (a ring) R is called 3-semiprime (semiprime) if, for all x∈R, xRx={0} implies x=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 near-ring is a υ-prime ideal of itself, where υ∈{0,1,2,3,e}. 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.
Several different generalizations of primeness for rings have been introduced for near-rings. In [6], Holcombe studied three different concepts of primeness, which he called 0-prime, 1-prime and 2-prime. In [5], Groenewald obtained further results for these and introduced further notion which he called 3-primeness. In [2], Booth, Groenewald and Veldsman gave another definition, called equiprimeness, or e-primeness. In [10], Veldsman made more studies on equiprime near-rings. In [1], Booth and Groenewald gave an element-wise characterization of the radical associated with ν -primeness for ν=1,2,3,e.
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) near-rings 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={0} implies U={0} or V={0}, 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 r2≠0 for all r∈R−{0}, 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.
2. On prime near-rings
Let R be a near-ring. It is clear that R is without zero divisors if and only if AB={0} implies A={0} or B={0}, where A and B are non-empty subsets of R. This observation gives us a hint of a new definition.
Definition 2.1. Let R be a non-zero near-ring.
(1) Let A be a non-empty subset of R. We say that A is a left zero-set divisor (a right zero-set divisor) of R if there exists a non-empty non-zero subset B of R such that AB={0} (BA={0}). We say that A is a zero-set divisor of R if A is a left or a right zero-set divisor of R.
(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={0} (BA={0}). 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.
(2) R is 0-prime if and only if R is without zero-ideal divisors.
(3) R is 1-prime if and only if R is without zero-right ideal divisors.
(4) R is 2-prime if and only if R is without zero-right R-subgroup 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:
(ii)aU={0} implies a=0 or U={0}, where a∈R and U is a semigroup left ideal of R.
(iii)AU={0} implies A={0} or U={0}, where A is a non-empty subset of R and U is a semigroup left ideal of R.
(iv)UV={0} implies U={0} or V={0}, where U and V are semigroup left ideals of R.
Proof. (i) implies (ii), (ii) implies (iii) and (iii) implies (iv) are clear.
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={0}. Thus, RxRy={0}. But Rx and Ry are semigroup left ideals of R, so Rx={0} or Ry={0} by (iv). Hence, R{0,x}={0} or R{0,y}={0} and either {0,x} or {0,y} is a semigroup left ideal of R. But R is also a semigroup left ideal of R. Thus, {0,x}=0, {0,y}={0} or R={0} 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:
(ii)Ua={0} implies a=0 or U={0}, where a∈R and U is a semigroup right ideal of R.
(iii)UA={0} implies U={0} or, A={0} where U is a semigroup right ideal of R and A is a non-empty subset of R.
(iv)UV={0} implies U={0} orV={0}, where U and V are semigroup right ideals of R.
(v)UV={0} implies U={0} orV={0}, where U is a semigroup right ideal of R and V is a semigroup left ideal of R.
Now, we can add (5) to Remark 2.1:(5)R is 3-prime if and only if UV={0} implies U={0} or V={0}, where U and V are semigroup left ideals of R if and only if R is without zero-semigroup left ideal divisors.
Since any ring is a zero-symmetric near-ring, we have the following result:
Corollary 2.3. 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:
(ii)aU={0} implies a=0, where a∈U and U is a semigroup left ideal of R.
(iii)Ua={0} implies a=0, where a∈U and U is a semigroup right ideal of R.
(iv)U2={0} implies U={0}, where U is a semigroup left ideal of R.
(v)U2={0} implies U={0}, where U is a semigroup right ideal of R.
Proof. (i) implies (ii). Suppose (i) holds. Let U be a semigroup left ideal of R such that aU={0}, where a∈U. Then for all v∈U, we have aRv={0}. Thus, aRa={0} and a=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 U2={0}, where U is a semigroup right ideal of R. So uRu={0} for all u∈U and hence RuRu={0}. But Ru is a semigroup left ideal of R. So Ru={0} for all u∈U by (iv). So {0,u} is a semigroup left ideal of R and {0,u}{0,u}={0} for all u∈U. So u=0 by (iv) and hence U={0}.
(v) implies (i). Suppose that (v) holds and that xRx={0} for some x∈R. Thus, xRxR={0}. But xR is a semigroup right ideal of R, so xR={0} by (v). Hence, {0,x}{0,x}={0}. But {0,x} is a semigroup right ideal of R. Thus, {0,x}={0} by (v) and hence x=0. So R is 3-semiprime and (v) implies (i). ■
Corollary 2.5. A ring R is semiprime if and only if U2={0} implies U={0}, where U is a semigroup right (left) ideal of R.
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:
(ii)aU={0} implies a=0, where a∈U and U is a semigroup left ideal of R.
(iii)U2={0} implies U={0}, where U is a semigroup left ideal of R.
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 (R,+) be the Klein’s four group {0,a,b,c}. Then it is an abelian group such that x+x=0 for all x∈R and x+y=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, {0,a} and {0,a,b}. So R satisfies the conditions “UV={0} implies U={0} or V={0}, where U and V are semigroup right ideals of R” and “U2={0} implies U={0}, where U is a semigroup right ideal of R”. But R is not 3-semiprime as bRb={0}. From Theorem 2.6, we can deduce that there is a non-zero semigroup left ideal V of R such that V2={0} and υV={0}, where υ∈V−{0}. It is easy to find out that V={0,b} and v=b.From the above example, observe that
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 r2≠0 for all r∈R−{0}, then R is 3-semiprime.
Proof. Suppose there exists a non-zero semigroup left ideal U of R such that aU={0},
where a∈U. That means a2=0. By hypothesis, a=0 and hence R is 3-semiprime. ■
Example 2. Let R=ℤ6. Then R is semiprime since r2≠0 for all r∈R−{0}.
Example 3. Let R={0,2,4,6,8,10,12} the subring of ℤ14. Then R is semiprime since r2≠0 for all r∈R−{0}.
The converse of Corollary 2.7 is not true as the following example shows.
Example 4.Let R=M2(ℤ2). 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 r2≠0 for all r∈R−{0} 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={0}. Then R is 3-prime if and only if R is 2-prime.
Proof. Suppose that xRy={0}. Thus, xRyR={0}. But xR and yR are right R-subgroups of R. So xR={0} or yR={0} as R is 2-prime. Hence, {0,x}R={0} or {0,y}R={0} and then either {0,x} or {0,y} is a right R-subgroup of R. But R is also a right R-subgroup of R. Thus, {0,x}=0, {0,y}={0} or R={0}. Hence, x=0 or y=0 and R is 3-prime. ■
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={0} for some x,y∈R. So xRyR={0} and hence xR={0} or yR={0}. So AR={0} or BR={0}, where A={nx|n∈ℤ} and B={ny|n∈ℤ}. So A and B are right R-subgroups of R and hence A={0} or B={0}. Therefore, x=0 or y=0 and R is 3-prime. ■
3. 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.
Theorem 3.1.Let R be a ring. Then the following statements are equivalent:
(ii)for any 0≠a,x,y∈R, xsa=yra for all s,r∈R−{0} implies x= 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=yca for all c∈R implies x= y.
(i) implies (ii) is clear.
(ii) implies (i). Suppose (ii) holds. If for all c∈R, xca=yca for 0≠a,x,y∈R, then (x−y)ca=0=0ra for all c,r∈R. So x=y by (ii). ■
Part (ii) enhances the following definition for near-rings.
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=yra for all s,r∈R−{0} implies x= 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=yrc=c for all s,r∈R−{0}, a contradiction with the hypothesis. So R is zero-symmetric. Now, suppose xRy={0} for some x,y∈R. So xcy=0 for all c∈R. If y≠0, then xcy=0ry for all c,r∈R. So x=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.
Example 5. Let (R,+) be the cyclic group ℤ5 and define the multiplication on R as follows:
So R is an abelian near-ring which is not a ring (as (1+1)2=3≠4=2+2=(1)2+(1)2). Clearly that R is without zero divisors. Hence, R is 3-prime. R is not equiprime. Indeed, 1c1=3c1=c1 for all c∈R. But if 0≠a,x,y∈R such that xsa=yra for all s,r∈R−{0}, then x= y. 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=yra for all s,r∈R−{0} and x,y,a are all non-zero, then from the table we can choose so,ro∈R−{0} to satisfy that xso=1 and yro=2. Hence, a=2a which implies that a=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 3-prime. Taking two non-zero elements x and y such that x≠y, we have xsx=yrx=x for all s,r∈R−{0}. 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:
Remark 3.1.Observe that:
(i)It is well-known that Mo(G) is e-prime (see [10]) and hence K-prime. Observe that it has zero divisors.
(ii)Since M(G) is not zero-symmetric, so it is not K-prime (and hence not e-prime), but it has zero divisors.
(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=yca for all c∈R, we have that x=y by choosing c=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 (K-primeness) 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:
4. 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 {0} 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
for all x,y∈R and r,s∈R−P. So x−y∈P and hence x−0=x∈P for all x∈R. Thus, P=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=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
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.
Proof.(i) implies (ii). Suppose (i) holds. Let U be a semigroup left ideal of R and B be a non-empty subset of R such that BU⊆P. If B⊈P, then there exists b∈B−P such that bRu⊆P for all u∈U. Thus, U⊆P by (i).
(ii) implies (iii) is clear.
(iii) implies (i). To prove it, we will use the contradiction. Suppose that (iii) holds and P is not a 3-prime ideal. So there exist x,y R−P such that xRy⊆P. Thus, RxRy⊆P. So Rx⊆P or Ry⊆P by (iii). Hence, R(P ∪ {x})⊆P or R(P ∪ {y})⊆P and then P ∪ {x} or P ∪ {y} is a semigroup left ideal of R. But R itself is also a semigroup left ideal of R. Thus, P ∪ {x}⊆P, P ∪ {y}⊆P or R⊆P by (iii), a contradiction with that x,y∈R−P. So P is 3-prime and (iii) implies (i). ■
Remark 4.1.From Theorem 4.2, a new characterization of 3-prime ideals can be written as follows:
(*) P is a 3-prime ideal of R if for every two semigroup left ideals A and B of R, AB⊆P implies A⊆P or B⊆P.
Using Theorem 4.2 and its proof, we can prove the following result which generalizes Theorem 2.2 for 3-prime ideals.
Theorem 4.3.Let R be a zero-symmetric near-ring and P an ideal of R. Then the following statements are equivalent:
(i)P is a 3-prime ideal of R.
(ii)UB⊆P implies U⊆P or B⊆P, where U is a semigroup right ideal of R and B is a non-empty subset of R.
(iii)UV⊆P implies U⊆P or V⊆P, where U and V are semigroup right ideals of R.
We cannot eliminate the condition “zero-symmetric” in Theorem 4.3 as the following example shows:
Example 7.Observe that {0} is not a 3-prime ideal in Example 1 although it satisfies the condition “If UV⊆{0}, then U⊆{0} or V⊆{0}, where U and V are semigroup right ideals of R”. This shows that “zero-symmetric” in Theorem 4.3 is not redundant.
Now, we would like to generalize Definition 2.1.
Definition 4.2.Let R be a near-ring with an ideal I.
(i)Let A be a non-empty subset of R. We say that A is a left I set-attractor (a right I set-attractor) of R if there exists a non-empty subset B of R and B⊈I such that AB⊆I (BA⊆I). We say that A is an I set-attractor of R if A is a left or a right I set-attractor of R.
(ii)Let A be an ideal of R. We say that A is a left I ideal-attractor (a right I ideal-attractor) of R if there exists an ideal B of R and B⊈I such that AB⊆I (BA⊆I). We say that A is an I ideal-attractor of R if A is a left or a right I ideal-attractor of R.
We can do the same definitions if A is a left (right) ideal of R, a left (right, two-sided) R-subgroup of R, a semigroup ideal of R or a semigroup left (right) ideal of R.
Example 8.Let R be a near-ring with an ideal I≠R. Any non-empty subset of I is a right I set-attractor of R and hence an I set-attractor of R. In particular, I is an I set-attractor of R. Also, if there exist an ideal (a left (right) ideal, a left R-subgroup, a semigroup left ideal) B of R such that B⊈I, then I is an I ideal-attractor (I left (right) ideal-attractor, I left R-subgroup-attractor, I semigroup left ideal-attractor) of R.
Definition 4.3.Let R be a near-ring with an ideal P. If A is a P set-attractor (P ideal-attractor, etc.) of R, then we say that A is an internal P set-attractor (P ideal-attractor, etc.) 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 ideal-attractor, 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 {0}-set attractors.
(iii)Let R be the ring ℤ4. Take P to be the ideal {0,2}. Then R is without external P set-attractors.
(iv)Let R be the ring ℤ6. Take P to be the ideal {0}. Then {2}, {3} and {4} are external P set-attractors and {0} 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 {a}{b}⊆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. ■
Remark 4.2. (i)If I={0} in Definition 4.2, then we have Definition 2.1.
(ii)From the above two definitions, Theorem 4.2 and 4.4, we can rewrite the statements of different kinds of prime ideals as follows:
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.