韓國數學敎育學會 Korean Soc. Math. Ed. <全國數學敎育硏究大會 프로시딩> Proceedings of National Meeting of Math. Ed.
제 44회. 2010. 4. 17-20 The 44th. Apr. 2010. 17-20
17
STRONG TYPES OF REGULAR NEAR-RINGS
Yong Uk Cho (Silla University)
1. Introduction
A near-ring is an algebraic system ⋅ with two binary operations and ⋅ such that is a group (not necessarily abelian) with neutral element , ⋅ is a semigroup and for all in . If has a unity , then is called
unital. A near-ring with the extra axiom for all ∈ is said to be zero symmetric. An element in is called distributive if for all and in .
Mason [1] introduced the notion of left regularity and characterized left regular zero-symmetric unital near-rings. Also, several authors ([1], [2], [4] etc.) studied them.
We will use the following notations: Given a near-ring , ∈ which is called the zero symmetric part of , ∈ which is called the constant part of .
Obviously, we see that and are subnear-rings of , but is a semigroup under multiplication. Clearly, near-ring is zero symmetric, in case also, in case , is called a constant near-ring.
For other notations and basic results, we shall refer to Pilz [3].
2. Results
A near-ring is called (Von Neumann) regular if for any element ∈, there exists an element in such that . Such an element a is called regular.
A near-ring is said to be left regular if, for each ∈, there exists ∈ such that . Right regularity is defined in a symmetric way. Also, we can generalize these concepts as following.
A near-ring is called strongly left regular if is left regular and regular, similarly, we can define strongly right regular. A strongly left regular and strongly right regular near-ring is called strongly regular near-ring. Also, the concepts of left, strongly left, strongly right and
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strong regularities are all equivalent conditions.
We say that is reduced if has no nonzero nilpotent elements, that is, for each in , , for some positive integer implies . In ring theory, McCoy proved that is reduced if and only if for each in , implies . A near-ring
is said to be
strongly reduced if, for ∈, ∈
implies ∈, that is
implies . Obviously is strongly reduced if and only if, for ∈ and any positive integer , ∈
implies ∈.
Obviously, we get the following examples by the concept of strong re-ducibility. Examples 2.1. (1) Every strongly regular near-ring is strongly reduced. (2) Every right regular near-ring is strongly reduced.
(3) Every commutative integral near-ring is strongly reduced.
Lemma 2.2. Let be a strongly reduced near-ring. Then we have the following conditions. (1) If for any ∈ with ∈, then ∈, and ∀∈, ∈.
Furthermore, ∈ implies ∈, for each positive integer .
(2) If for any ∈ with , then . Moreover, implies
, for any positive integer .
Proof. (1) Suppose that ∈. Then
∈. Since is strongly reduced, we have ∈.
Next, we see that ∈ for each ∈, whence ∈. By the strong reducibility of , we obtain ∈ for each ∈. Also, since ∈, we obtain ∈ for each ∈.
Furthermore, assume that ∈
. Then using the first part of this (1), ∈. Since is strongly reduced, we see ∈.
(2) Assume that . Then ∈ by (1). Hence ∈. Hence ∈. Therefore we obtain that . Moreover, suppose that . Then ∈ by the last part of (1), so that .
Lemma 2.3. Let be a strongly reduced near-ring. If for any ∈ with and , then .
STRONG TYPES OF REGULAR NEAR-RINGS 19
Proof. Suppose that for any ∈ with and . Then ∈
. Strong reducibility implies that ∈. Hence we obtain that .
Corollary 2.4. Every strongly reduced near-ring is reduced.
By Reddy and Murty [4], we say that a near-ring has the property (*) if it satisfies the conditions:
(i) for any ∈, implies . (ii) for ∈, implies .
Here, clearly we see that strong reducibility is equivalent to the condition (ii) and strong reducibility implies condition (i) by Lemma 2 (2).
According to the Lemmas 2.2 and 2.3, we have the following valuable corollary.
Corollary 2.5. Let be a left (or right) regular near-ring. If for any ∈ with ,
then , for all positive integer . In particular, .
Clearly, if is a zero-symmetric near-ring, then is strongly reduced if and only if is reduced.
Proposition 2.6. The following statements are equivalent for a near-ring : (1) is strongly reduced.
(2) For ∈, implies .
(3) If for ∈ and some nonnegative integer , then .
Proof. (1) ⇒ (2). Assume that . Then , whence ∈ by Lemma 2.2 (2). Then
. Again by Lemma 2.2 (2),
∈. Hence ∈. This implies ∈. Hence
. (2) ⇒ (1). Assume ∈
. Then . By hypothesis, this implies ∈. (1) ⇒ (3). Suppose for some ≥ . Then . Hence by Lemma 2.2 (2), and so ∈
by Lemma 2.2 (1). Since is strongly reduced, we have ∈. Then , that is . Now
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∈
. Hence ∈ by Lemma 2.2 (1), and so ∈. Therefore .
(3) ⇒ (2). This is obvious.
References
Mason, G. (1998). A note on strong forms of regularity for near-rings, Indian J. of Math. 40(2), pp.149-153.
Murty, C. V. L. N. (1984). Generalized near-fields, Proc. Edinburgh Math. Soc. 27, pp.21-24. Pilz, G. (1983). Near-Rings, North-Holland Publishing Company, Amsterdam, New York, Oxford. Reddy, Y. V., & Murty, C. V. L. N. (1984). On strongly regular near-rings, Proc. Edinburgh
