1 Introduction
Throughout this paper, by a near-ring N we mean a zero-symmetric near-ring with identity 1. For basic definitions in near-rings one may refer to [1]. In 1970 W.L.M. Holcombe introduced the notion of (0, 1, 2)-prime ideals of a near-ring [2]. In 1977 G. Pilz introduced the notion of prime ideals of a near-ring [1]. In 1988 N.J. Groenewald introduced the notion of completely (semi) prime ideals of a near-ring [3]. In 1991 N.J. Groenewald introduced the notion of 3-(semi) prime ideals of a near-ring [4]. In [5] D.D. Anderson and E. Smith defined weakly prime ideals in commutative rings; an ideal P of a ring R is weakly prime if 0 ≠ ∈ ab P implies a P ∈ or b P ∈ .
In 2012 H.H. Abbass and S.M. Ibrahem introduced the concept of a completely semi prime ideal with respect to an element of a near-ring and the completely prime ideals in near-rings with respect to an element of a near-ring [6].
In this investigation we studied completely prime, weakly completely prime, quasi completely prime and weakly quasi completely prime ideals in near-rings. Some characterizations of completely prime and weakly completely prime ideals were obtained. Moreover, we investigated relationships between completely prime and weakly completely prime ideals in near-rings. Finally, we
obtained necessary and sufficient conditions for a weakly completely prime ideal to be a completely prime ideal.
2 Basic Results
In this section we refer to [1,4,6,7] for some elementary aspects and quote a number of theorems and lemmas that are essential to step up this study. For more details we refer to the papers in the references.
Definition 2.1 [6] A near-ring is a triple (N, , + ⋅) of a nonempty set N together with two binary operations "+" and " ⋅ " (called addition and multiplication respectively) defined on N such that the following holds:
- (i) (N,+) is a group.
- (ii) (N,⋅) is a semigroup.
- (iii) (b c a ba ca + =+ ) , for all abc N ,, . ∈
Definition 2.2 [1] An ideal P of a near-ring N is called a completely prime ideal if for ab N , ∈ such that ab P ∈ implies that a P ∈ or b P ∈ .
Definition 2.3 An ideal P of a near-ring N is called a weakly completely prime ideal if for ab N , ∈ such that 0 ≠ ∈ ab P implies that a P ∈ or b P ∈ .
Clearly, every completely prime ideal is weakly completely prime and {0} is always a weakly completely prime ideal of N. The following example shows that a weakly completely prime ideal need not be a completely prime ideal in general.
Example 2.4 [4] Let N abcd ={0, , , , ,1,2,3}. Define addition and multiplication in N as follows:
| + | 0 | 1 | 2 | 3 | a | b | c | c |
| 0 | 0 | 1 | 2 | 3 | a | b | c | d |
| 1 | 1 | 2 | 3 | 0 | d | c | a | b |
| 2 | 2 | 3 | 0 | 1 | b | a | d | c |
| 3 | 3 | 0 | 1 | 2 | c | d | b | a |
| a | a | d | b | c | 2 | 0 | 1 | 3 |
| b | b | c | a | d | 0 | 2 | 3 | 1 |
| c | c | a | d | b | 1 | 3 | 0 | 2 |
| d | d | b | c | a | 3 | 1 | 2 | 0 |
| ⋅ | 0 | 1 | 2 | 3 | a | b | c | d |
|---|---|---|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 2 | 3 | a | b | c | d |
| 2 | 0 | 2 | 0 | 2 | 2 | 2 | 0 | 0 |
| 3 | 0 | 3 | 2 | 1 | b | a | c | d |
| a | 0 | a | 2 | b | a | b | c | d |
| b | 0 | b | 2 | a | b | a | c | d |
| c | 0 | c | 0 | c | 0 | 0 | 0 | 0 |
| d | 0 | d | 0 | d | 2 | 2 | 0 | 0 |
Then (N, , + ⋅) is a near-ring but (N, , + ⋅) is not a ring. Here {0, c} is a weakly completely prime ideal but not completely prime, since 2 2 0 0, c . ⋅=∈{ }
Lemma 2.5 [7] Let A be an ideal of (N,,. + ⋅) Then N A/ is a near-ring under the operations: For all ab N , ∈
\[(a+A)+(b+A)=(a+b)+A\] and \((a+A)(b+A)=(ab)+A\).
Lemma 2.6 [7] Let A and B be ideals of (N,,. + ⋅) Then
\[(A+B)/A \approx B/(A \cap B).\]
Furthermore, if A B ⊆ , then (NA BA NB / / / /. ) ( ) ≈
3 Main Results
We start with the following theorem that gives a relation between completely prime and weakly completely prime ideals in a near-ring. Our starting point is the following lemma:
Lemma 3.1 Let N be a near-ring and let A be a left ideal of N . Then ( A B: ) is a left ideal in N , where ( A B n N nB A : :. ) =∈ ⊆ { }
Proof. Let N be a near-ring and let A be a left ideal of N . Suppose that n N ∈ and mn A B , :. ∈( ) Then mB A ⊆ and nB A ⊆ so that
\[(n-m)B = nB - mB \subseteq A.\]
Therefore n m AB − ∈( : .) For a AB ∈( : ) and n N ∈ ,
\[(n+a-n)B = nB + aB - nB\] \[\subseteq nB + A - nB\] \[\subseteq A\] since A is a left ideal of N . Therefore, n a n AB +−∈( : .) Thus ( A B: ) is a normal subgroup of N . Let mn Na A B , , :. ∈ ∈( ) Then
\[(m(n-a)-mn)B = (m(n-a))B - (mn)B\] \[= m((n-a)B) - (mn)B\] \[= m(nB-aB) - (mn)B\] \[= m(nB-aB) - (mn)B\] \[\subseteq A\]
Thus m n a mn A B ( −− ∈ ) ( : .) Hence ( A B: ) is a left ideal in N.
Definition 3.2 [1] A left ideal P of a near-ring N is called a quasi completely prime ideal if for ab N , ∈ such that ab P , ∈ implies that a P ∈ or b P ∈ .
Definition 3.3 A left ideal P of a near-ring N is called a weakly quasi completely prime ideal if for ab N , ∈ such that 0 ≠ ∈ ab P implies that a P ∈ or b P ∈ .
Theorem 3.4 Let N be a near-ring, and let A be an ideal of N . If A is a weakly quasi completely prime (quasi completely prime) ideal of N , then ( A B: ) is a weakly quasi completely prime (quasi completely prime) ideal in N , where B A ⊄ .
Proof. Let N be a near-ring, and let A be a weakly quasi completely prime ideal of N . Suppose that 0 : ≠ ∈ mn A B ( ) and m AB ∉( : .) Then
\[0 \neq m(nB) = (mn)B \subseteq A\].
By the definition of weakly quasi completely prime ideal we get m A ∈ or nB A ⊆ so that n AB ∈( : .) Hence ( A B: ) is a weakly quasi completely prime ideal in N .
Corollary 3.5 Let N be a near-ring and let A be a weakly quasi completely prime (quasi completely prime) ideal of N . Then ( A m: ) is a weakly quasi completely prime (quasi completely prime) ideal in N , where mNA ∈ − .
Proof. This follows from Theorem 3.4.
Theorem 3.6 Let N be a near-ring and let P be an ideal of N . If P is a weakly completely prime ideal that is not completely prime, then 2 P = 0.
Proof. Let N be a near-ring. Suppose that 2 P ≠ 0 we show that P is a completely prime ideal in N. Let ab P ∈ , where ab N , . ∈ If ab ≠ 0, then either
\[a \in P\] or \(b \in P\)
since P is a weakly completely prime ideal. So suppose that ab = 0. If Pb ≠ 0, then there is an element p′ of P such that p b′ ≠ 0, so that
\[0 \neq p'b = p'b + 0 = p'b + ab = (p' + a)b \in P,\] and hence P as a weakly completely prime ideal gives either p aP ′ + ∈ or b P ∈ . As p aP ′ + ∈ and p P ′∈ we have either a P ∈ or b P ∈ . So we can assume that Pb = 0. Similarly, we can assume that Pa = 0. Since 2 P ≠ 0, there exist cd P , ∈ such that cd 0. ≠ Then
\[0 \neq (a+c)(b+d) \in P,\] so either a cP , + ∈ or b dP , + ∈ and hence either a P ∈ or b P ∈ . Thus P is a completely prime ideal. Clearly, 2 0 . ⊆ P Hence, 2 P = 0, as required.
Corollary 3.7 Let N be a near-ring and let P an ideal of N . If 2 P ≠ 0, then P is completely prime if and only if P is weakly completely prime.
Proof. This follows from Theorem 3.6.
Lemma 3.8 Let N be a near-ring, and let P be a proper ideal of N . If P is a weakly completely prime ideal of N , then
\[(P:Na) = P \cup (0:Na),\] where aNP ∈ − .
Proof. Let N be a near-ring, and let P be a weakly completely prime ideal of N . Clearly,
\[P \cup (0:Na) \subseteq (P:Na).\]
For the other inclusion, suppose that m P Na ∈( : ,) so that
\[m(Na) \subseteq P\].
If 0 ≠ m Na ( ) and P is a weakly completely prime ideal of N then Na P ⊆ . If 0 , = m Na ( ) then m Na ∈(0: .) So we have the equality.
Corollary 3.9 Let N be a near-ring and let P be a proper ideal of N . If P is a weakly completely prime ideal of N, then (Pa P a : 0: , ) = ∪ ( ) where aNP ∈ − .
Proof. This follows from Lemma 3.8.
Corollary 3.10 Let N be a near-ring with identity, P be a proper ideal of N and aNP ∈ − . If (P Na P Na : 0: ) = ∪( ) and ( : ) , P Na P = then ( : ) (0 : ). P Na Na =
Proof. This follows from Lemma 3.8.
Theorem 3.11 Let N be a near-ring with identity and let P be a proper ideal of N. If ( : ) Pn P = or ( : ) (0 : ), Pn n = then P is a weakly completely prime ideal of N, where nNP ∈ − .
Proof. Let N be a near-ring with identity and let P be a proper ideal of N. Suppose that Let 0 , ≠ ∈ mn P where mNP ∈ − . Then
\[m \in (P:n) = P \cup (0:n)\] by Corollary 3.10, hence m P ∈ since mn ≠ 0 , as required.
Theorem 3.12 Let 1 2 NNN = × , where each Ni is a near-ring with identity. If P is a weakly completely prime (completely prime) ideal of 1 N , then P N× 2 is a weakly completely prime (completely prime) ideal of N.
Proof. Suppose that 1 2 NNN = × , where each Ni is a near-ring with identity and P is a weakly completely prime ideal of 1 N . Let
\[0 \neq (a,b)(c,d) = (ac,bd) \in P \times N_2,\] where (ab cd N , ,, ) ( )∈ so either a P ∈ or c P ∈ since P is weakly completely prime. It follows that either
\[(a,b) \in P \times N_2 \text{ or } (c,d) \in P \times N_2\]
By the definition of weakly completely prime ideal, we have \(P \times N_2\) is a weakly completely prime ideal of N.
Corollary 3.13 Let \(N = N_1 \times N_2\), where each \(N_i\) is a near-ring with identity. If P is a weakly completely prime (completely prime) ideal of \(N_2\), then \(N_1 \times P\) is a weakly completely prime (completely prime) ideal of N.
Proof. This follows from Lemma 3.12.
Corollary 3.14 Let \(N = \prod_{i=1}^{n} N_i\), where each \(N_i\) is a near-ring with identity. If P is a weakly completely prime (completely prime) ideal of \(N_j\), then \(N_1 \times N_2 \times ... \times P_j \times N_{j+1} \times ... \times N_n\) is a weakly completely prime (completely prime) ideal of N.
Proof. This follows from Theorem 3.12 and Corollary 3.13.
Theorem 3.15 Let \(N = N_1 \times N_2\), where each \(N_i\) is a near-ring with identity. If P is a weakly completely prime ideal of N, then either P = 0 or P is completely prime.
Proof. Let \(N = N_1 \times N_2\), where each \(N_i\) is a near-ring with identity and let \(P = P_1 \times N_2\) be a weakly completely prime ideal of N. We can assume that \(P \neq 0\). So there is an element (a,b) of P with \((a,b) \neq (0,0)\). Then
\[(0,0) \neq (a,1)(1,b) \in P\], gives either
\[(a,1) \in P\] or \((1,b) \in P = P_1 \times N_2\).
If \((a,1) \in P\), then \(P = P_1 \times N_2\). We will show that \(P_1\) is completely prime hence P is weakly completely prime by Theorem 3.12. Let \(cd \in P_1\), where \(c,d \in N_1\). Then
\[(0,0) \neq (c,1)(d,1) = (cd,1) \in P\], so either \((c,1) \in P\) or \((d,1) \in P\) and hence either \(c \in P_1\) or \(d \in P_1\). By a similar argument, \(N_1 \times P_2\) is completely prime.
Proposition 3.16 Let A P ⊆ be proper ideals of a near-ring N. Then the following holds:
- (i) If P is weakly completely prime (completely prime), then P A/ is weakly completely prime (completely prime).
- (ii) If A and P A/ are weakly completely prime (completely prime), then P is weakly completely prime (completely prime).
Proof. (i) Let 0 ≠ + + = +∈ (a A b A ab A P A )( ) / , where ab N , , ∈ so ab P ∈ . If ab A = ∈0 , then
\[(a+A)(b+A)=0,\] a contradiction. So if P is weakly completely prime, then either a P ∈ or b P ∈ , hence either a A PA + ∈ / or b A PA + ∈ / , as required.
(ii) Let 0 , ≠ ∈ ab P where ab N , , ∈ so (a Ab A PA + +∈ )( ) / . For ab A ∈ , if A is weakly completely prime, then either
\[a \in A \subseteq P\] or \(b \in A \subseteq P\).
So we may assume that ab A ∉ . Then either a A PA + ∈ / or b A PA + ∈ / . It follows that either a P ∈ or b P ∈ as needed.
Theorem 3.17 Let P and Q be weakly completely prime ideals of a near-ring N that are not completely prime. Then P Q+ is a weakly completely prime ideal of N.
Proof. Since (PQ QQ P Q +≈∩ ) // , ( ) we get that (PQ Q + ) / is weakly completely prime by Proposition 3.16 (i). Now the assertion follows from Proposition 3.16 (ii).
Acknowledgements
The authors are very grateful to the anonymous referees for their stimulating comments and improving the presentation of this paper.