Prime ideals and radical of ideals in near- rings

The most interesting and active field of current research in mathematics is theory of near- ring due to its wide application in coding theory, group theory, geometry, cryptography and block- designing etc. Near rings are one of the generalized structure of rings. In this paper we discuss the concepts of ideals and radicals in near- rings also the concepts of m- system and sp- system in context of near- rings. Also, the concepts of m*- system, m*- sequences and prime radicals in near rings.

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Introduction

The most interesting and active field of current research in mathematics is theory of near-ring due to its wide application in coding theory, group theory, geometry, cryptography and blockdesigning etc. Near rings are one of the generalized structure of rings, The study and research on near-rings is very systematic and continuous. Near-rings around in all directions of mathematics and continuous research is being conducted, which show that their structure has power and beauty all its own. Modern ring theory has very active mathematical discipline and studying rings in their own right. In 1930 Wieland studied near-rings. In this paper we discuss the concepts of ideals and radicals in near-rings also the concepts of 𝑚system and 𝑠𝑝system in context of near-rings. Also, the concepts of 𝑚 * -system, 𝑚 * -sequences and prime radicals in near rings. Definition 1.1: A system (𝑁, +, . ) is called a near-ring if a) (𝑁, +)is a group (not necessarily abelian). b) (𝑁, . ) is a semi group c) Multiplication is either left distributive or right distributive over addition. That is (𝑛 1 + 𝑛 2 ). 𝑛 3 = 𝑛 1 . 𝑛 3 + 𝑛 2 . 𝑛 3 , Or, 𝑛 1 . (𝑛 2 + 𝑛 3 ) = 𝑛 1 . 𝑛 2 + 𝑛 1 . 𝑛 3 , for 𝑎𝑙𝑙 𝑛 1 , 𝑛 2 , 𝑛 3 ∈ 𝑁.

Example 1: Every ring is a near-ring.

Definition 1.2:

A subset 𝑆 of a near-ring 𝑁 𝑖s called an 𝑚system if for every 𝑎, 𝑏 ∈ 𝑆, there exist 𝑎 1 ∈< 𝑎 > and 𝑏 1 ∈< 𝑏 > such that 𝑎 1 , 𝑏 1 ∈ 𝑆.

Example 2: Let 𝑀= ℤ 4 = {0, 1, 2, 3}𝑚𝑜𝑑(4), 𝑆 = {1, 2}𝑚𝑜𝑑(4). Clearly, 𝑆 ⊆ ℤ 4, 𝑆 is an 𝑚system.

Theorem 1.3: An ideal 𝑃 in 𝑁 is a prime ideal if and only if its complement 𝑃 𝐶 is an 𝑚system.

Proof: Suppose that P is a prime ideal then to prove that 𝑃 𝐶 is an 𝑚system. Let 𝑎, 𝑏 ∈ 𝑃 𝐶 ⇒ < 𝑎 >⊆ 𝑃 𝐶 𝑎𝑛𝑑 < 𝑏 >⊆ 𝑃 𝐶 ⇒ ∃ 𝑎 1 ∈< 𝑎 >⊆ 𝑃 𝐶 𝑎𝑛𝑑 𝑏 1 ∈< 𝑏 >⊆ 𝑃 𝐶 ⇒ < 𝑎 >⊈ 𝑃 𝑎𝑛𝑑 < 𝑏 >⊈ 𝑃 Hence by the definition of prime ideal we have, < 𝑎 >< 𝑏 >⊈ 𝑃.

So, 𝑎 1 𝑏 1 ∉ 𝑃 ⇒ 𝑎 1 𝑏 1 ∈ 𝑃 𝐶 . Hence 𝑃 𝐶 is an 𝑚system. Conversely, suppose that 𝑃 𝐶 is an 𝑚system, then to prove that 𝑃 is a prime ideal, Suppose that 𝐵 and 𝐷 be two ideals such that 𝐵𝐷 ⊆ 𝑃 then to prove either 𝐵 ⊆ 𝑃 or 𝐷 ⊆ 𝑃. Suppose that neither 𝐵 ⊆ 𝑃 nor 𝐷 ⊆ 𝑃. ⇒ 𝐵 ⊆ 𝑃 𝐶 or𝐷 ⊆ 𝑃 𝐶 then as 𝑃 𝐶 is an 𝑚system, so for every 𝑏 ∈ 𝐵 ⊆ 𝑃 𝐶 and 𝑑 ∈ 𝐷 ⊆ 𝑃 𝐶 ∃ 𝑏 ′ ∈< 𝑏 >⊆ 𝐵 𝑎𝑛𝑑 𝑑 ′ ∈< 𝑑 >⊆ 𝐷 such that 𝑏 ′ 𝑑 ′ ∈ 𝑃 𝐶 ⇒ 𝑏 ′ 𝑑 ′ ∉ 𝑃 so 𝐵𝐷 ⊈ 𝑃 ⇒ 𝐵𝐷 ⊆ 𝑃 𝐶 , which is a contradiction. Hence 𝑃 is a prime ideal. Definition 1.4: An ideal 𝑆 of 𝑁 is said to be a semi-prime ideal if for any ideal 𝐽 of 𝑁, 𝐽 2 ⊆ 𝑆 ⇒ 𝐽 ⊆ 𝑆. Definition 1.5: A subset 𝑆 of a near-ring 𝑁 is said to be an 𝑠𝑝system if for any 𝑠 ∈ 𝑆, there exist 𝑠 1, 𝑠 2 ∈< 𝑠 > such that 𝑠 1 . 𝑠 2 ∈ 𝑆.

Example 3: For any 𝑛 ∈ 𝑁, the set {𝑛, 𝑛 2 , 𝑛 3 , … … . } is an 𝑠𝑝system. Theorem 1.6: 𝑃 be a semi-prime ideal in a near-ring 𝑁 if and only if 𝑁\𝑃 is an 𝑠𝑝system in 𝑁.

Proof: Suppose that 𝑃 is a semi-prime ideal of 𝑁. Then for any ideal 𝐽 of 𝑁 such that 𝐽 2 ⊆ 𝑃 ⇒ 𝐽 ⊆ 𝑃.

Claim: 𝑁\𝑃 is an 𝑠𝑝system in 𝑁. Let 𝑚 ∈ 𝑁\𝑃, then < 𝑚 >⊆ 𝑁\𝑃. Let 𝑚 ′ , 𝑚" ∈< 𝑚 > ⊆ 𝑁\𝑃 then to show 𝑚 ′ . 𝑚" ∈< 𝑚 > ⊆ 𝑁\𝑃. Now < 𝑚 >⊆ 𝑁\𝑃 ⇒ < 𝑚 >⊈ 𝑃. So by definition of semi-prime ideal it follows that< 𝑚 > 2 ⊈ 𝑃, hence 𝑚 ′ . 𝑚" ∉ 𝑃 ⇒ 𝑚 ′ . 𝑚" ∈< 𝑚 > ⊆ 𝑁\𝑃. Thus 𝑁\𝑃 is an 𝑠𝑝system. Conversely, let 𝑁\𝑃 be an 𝑠𝑝system that is for all 𝑛 ∈ 𝑁\𝑃, ∃ 𝑛 1 , 𝑛 2 ∈< 𝑛 > such that 𝑛 1 . 𝑛 2 ∈ 𝑁\𝑃.

Claim: 𝑃 is a semi-prime ideal. Let 𝐽 be an ideal of 𝑁 such that 𝐽 2 ⊆ 𝑃, we need to show 𝐽 ⊆ 𝑃. If it possible to show 𝐽 ⊈ 𝑃, then 𝐽 ⊆ 𝑁\𝑃, which is an 𝑠𝑝system. Therefore, ∀ 𝑛 ∈ 𝑁\𝑃, ∃ 𝑛 1 , 𝑛 2 ∈< 𝑛 > such that 𝑛 1 . 𝑛 2 ∈ 𝑁\𝑃. Hence 𝑛 1 . 𝑛 2 ∉ 𝑃 ⇒ 𝐽 2 ⊈ 𝑃, which is a contradiction. Hence 𝑃 is a semi-prime ideal.

Theorem 1.7: If 𝐼 is any semi-prime ideal in a near ring 𝑁, then 𝐼 is the intersection of all minimal prime ideals of 𝐼 in 𝑁.

Proposition 1.8:

If 𝐼 is a semi-prime ideal in a near ring 𝑁, then 𝐼 is the intersection of all the prime ideals containing 𝐼. Theorem 1.9: An ideal 𝐼 in a near ring 𝑁 is a semi-prime ideal in 𝑁 if and only if 𝓅(𝐼) = 𝐼.

Proof: Suppose that 𝓅(𝐼) = 𝐼, then to prove the ideal 𝐼 is a semi-prime ideal. Since 𝓅(𝐼) is the intersection of all prime ideals containing 𝐼 and the intersection of all prime ideals is a semi-prime ideal, so that 𝐼 is semi-prime ideal. Conversely, suppose that 𝐼 is semi-prime ideal, then to show 𝓅(𝐼) = 𝐼. Already we know that 𝐼 ⊆ 𝓅(𝐼). Now if it possible, suppose that 𝓅(𝐼) ⊈ 𝐼, then there exists 𝑎 ∈ 𝓅(𝐼) such that 𝑎 ∉ 𝐼. Since 𝐼 is a semi-prime ideal, so we have 𝑁\𝐼 is an 𝑠𝑝system. Hence there exists an 𝑚system 𝑀 in 𝑁 such that 𝑎 ∈ 𝑀 ⊆ 𝑁\𝐼. Now 𝑎 ∈ 𝓅(𝐼) and we know every 𝑚system which contains 𝐼. But 𝐼 ∩ 𝑁\𝐼 = ∅, and therefore 𝐼 ∩ 𝑀 is empty, which is a contradiction. Hence, 𝓅(𝐼) = 𝐼.

Radicals of an ideal in near-rings Definition:

A subset 𝑀 of 𝑁 is said to an 𝑚 * -system, if for any 𝑎, 𝑏 ∈ 𝑀, there exists an element 𝑏 1 ∈ (𝑏) such that 𝑎𝑏 1 ∈ 𝑀. Note: It is a generalization of 𝑚-system. Theorem 2.1: If 𝑀 𝑖s a non empty 𝑚 * -system and if 𝐼 is an ideal of 𝑁 such that 𝐼 ∩ 𝑀 = ∅, then there exists a prime ideal 𝑃 containing 𝐼 such that 𝑃 ∩ 𝑀 = ∅.

Proof: Suppose 𝑀 𝑖s a non empty 𝑚 * -system and if 𝐼 is an ideal of 𝑁 such that 𝐼 ∩ 𝑀 = ∅. We Consider 𝑁' = {𝐾| 𝐾 is an ideal of, 𝐼 ⊆ 𝐾, 𝐾 ∩ 𝑀 = ∅ }. We want to show 𝑁' contains a maximal element. Clearly 𝑁' is a partially ordered set with respect to the relation "⊆" and since 𝐼 ∈ 𝑁 ′ so 𝑁 ′ ≠ ∅. We show that every totally ordered subset 𝑇 of 𝑁' has an upper bound in 𝑁'. Let 𝑇 0 = ∪ 𝑇 𝑖 where 𝑇 𝑖 ∈ 𝑇, then obviously 𝑇 0 is an upper bound of 𝑁'. Now to show 𝑇 0 ∈ 𝑁′, since 𝑇 𝑖 ∈ 𝑇 ∈ 𝑁′. So 𝐼 ⊆ 𝑇 𝑖 ∀ 𝑖. This show that 𝐼 ⊆ 𝑇 0 , if 𝑥 ∈ 𝑇 0 , then 𝑥 ∈ 𝑇 𝑖 for some 𝑇 𝑖 ∈ 𝑇. But 𝑇 𝑖 is an ideal of 𝑁, so 𝑥𝑎 ∈ 𝑇 𝑖 ∈ 𝑇 0 and 𝑎(𝑏 + 𝑥) -𝑎𝑏 ∈ 𝑇 𝑖 ∈ 𝑇 0 for all 𝑥 ∈ 𝑇 𝑖 𝑎𝑛𝑑 𝑎, 𝑏 ∈ 𝑁. This shows that 𝑇 0 is an ideal of 𝑁. Also, if 𝑇 0 ∩ 𝑀 ≠ ∅, then there exists 𝑥 ∈ 𝑀 such that 𝑥 ∈ 𝑇 0 = ∪ 𝑇 𝑖 . This implies that 𝑥 ∈ 𝑇 𝑖 , for some 𝑇 𝑖 ∈ 𝑇. Thus𝑇 𝑖 ∩ 𝑀 ≠ ∅ for some 𝑖, which is a contradiction as 𝑇 𝑖 ∩ 𝑀 ≠ ∅. This shows that 𝑇 𝑖 ∩ 𝑀 = ∅. So, 𝑇 0 ∈ 𝑁′, then by using the Zorn's Lemma 𝑁' has a maximal element say '𝑃'. Since 𝑁\𝑃 is an 𝑚system then 𝑃 is a prime ideal.

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