Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
International Mathematical Forum, Vol. 6, 2011, no. 44, 2195 - 2199 Some Basic Properties of LA-Ring Muhammad Shah and Tariq Shah Department of Mathematics Quaid-i-Azam University, Islamabad, Pakistan shahmaths [email protected] [email protected] Abstract. In this note we discuss some basic properties of LA-ring. Preliminaries and introduction : An AG-groupoid is a groupoid satisfying the left invertive law: (ab)c = (cb)a. This structure is also known as left almost semigroup (LA-semigroup) in [4], left invertive groupoid in [2] while right modular groupoid in [1, line 35]. A medial is a groupoid satisfying the medial law: (ab)(cd) = (ac)(bd) [1, line 36] while a paramedial is a groupoid satisfying the paramedial law: (ab)(cd) = (db)(ca) [1, line 37]. An AG-groupoid S always satisfies the medial law: (ab)(cd) = (ac)(bd) [1, Lemma 1.1 (i)] while an AG-groupoid S with left identity e satisfies paramedial law: (ab)(cd) = (db)(ca) [1, Lemma 1.2 (ii)]. An AG-groupoid S with left identity e also satisfies a(bc) = b(ac) [6, Lemma 4, page 58]. An element a of an AG-groupoid S is called left cancellative if ax = ay ⇒ x = y for all x, y ∈ S. Similarly an element a of an AG-groupoid S is called right cancellative if xa = ya ⇒ x = y for all x, y ∈ S. An element a of an AGgroupoid S is called cancellative if it is both left and right cancellative. An AGgroupoid S is called left cancellative if every element of S is left cancellative. Similarly an AG-groupoid S is called right cancellative if every element of S is right cancellative and it is called cancellative if every element of S is both left and right cancellative. Kamran [3] and [5] extended the notion of AG-groupoid to LA-group (AGgroup). An AG-groupoid G is called an AG-group if there exists left identity −1 e ∈ G (that is ea = a for all a ∈ G), for all a ∈ G there exists a ∈ G such −1 that a a = aa−1 = e. Some basic properties of AG-groups can be found in [7]. In [9], S. M. Yusuf extended these to a non-associative structure with respect to both binary operations ‘+’ and ‘.’ namely left almost ring (LA-ring). By a left almost ring we mean a nonempty set R with at least two elements such that (R, +) is an AG-group and (R, .) is an AG-groupoid and both left and right distributive laws hold. An LA-ring (R, +, .) with left identity e is called M. Shah and T. Shah 2196 almost field if every nonzero element a of the ring has multiplicative inverse a−1 in R. A recent development can be found in T. Shah and I. Rehman [8]. LA-ring is not studied much and is in its initial form. In this note we are establishing some basic and structural facts of LA-ring which we hope will be proved useful for future research on LA-ring. Thus we study basic results such as if R is an LA-ring then R cannot be idempotent and also (a + b)2 = (b + a)2 ∀a, b ∈ R. If LA-ring R has left identity e then e + e 6= e, e + 0 6= e and e = (e + 0)2 . If R is a cancellative LA-ring with left identity e then e + e = 0 and thus a + a = 0 ∀a ∈ R. An interesting result is that if R is an LA-ring with left identity e then right distributivity implies left distributivity. The following example shows the existence of LA-ring. Example 1. An LA-ring of order 5 : + 0 1 2 3 4 0 0 4 3 2 1 1 1 0 4 3 2 2 2 1 0 4 3 3 3 2 1 0 4 4 4 3 2 1 0 · 0 1 2 3 4 0 0 0 0 0 0 1 0 1 2 3 4 2 0 2 4 1 3 3 0 3 1 4 2 4 0 4 3 2 1 We start by the following basic result:. Theorem 1. Let R be an LA-ring. Then (1) (a + b)(c + d) = (b + a)(d + c) ∀a, b, c, d ∈ R. Proof. (a + b)(c + d) = = = = = = a(c + d) + b(c + d) by right distributive law (ac + ad) + (bc + bd) by left distributive law (ac + bc) + (ad + bd) by medial law (bd + ad) + (bc + ac) by(a+b)+(c+d)=(d+c)+(b+a) (b + a)d + (b + a)c by right distributive law (b + a)(d + c) by left distributive law. Some basic properties of LA-ring 2197 Next we give a number of corollaries of Theorem 1 which present some distinguished features of LA-ring. Corollary 1. Let R be an LA-ring. Then ac = (a + 0)(c + 0) ∀a, c ∈ R. Proof. Take b = d = 0 in (1). Corollary 2. If LA-ring R has left identity e. Then e + e 6= e. Proof. Suppose e + e = e. Take a = b = e in (1) , we have c + d = d + c ∀c, d ∈ R ⇒ (R, +) is abelian group which is a contradiction. Hence e + e 6= e. Corollary 3. Let R be a cancellative LA-ring R with left identity e. Then e + e = 0. Proof. Suppose e + e 6= e. Take c = d = e in (1). Then by cancellation, we get a + b = b + a ∀a, b ∈ R ⇒ (R, +) is abelian group which is a contradiction. Hence e + e = 0. Corollary 4. Let R be a cancellative LA-ring R with left identity e. Then a + a = 0 ∀a ∈ R. Proof. By Corollary 2, e + e = 0. Now a + a = (e + e)a = 0a = 0 by right distributive law. Corollary 5. Let R be an LA-ring. Then (a + b)2 = (b + a)2 ∀a, b ∈ R. Proof. Take c = a, d = b in (1). Corollary 6. Let R be an LA-ring then a2 = (a + 0)2 ∀a ∈ R. Proof. Take b = 0 in Corollary 5. Corollary 7. Let R be an LA-ring R with left identity e, then e + 0 6= e. Proof. Suppose e + 0 = e. Let a is an arbitrary element of R. Now a = ea = (e + 0)a = e(a + 0) = a + 0 by Theorem 1 ⇒ (R, +) is abelian group, a contradiction. Hence e + 0 6= e. Corollary 8. Let R be an LA-ring R with left identity e. Then e = (e + 0)2 . Proof. Take a = e in Corollary 6. Recall the definition of an idempotent ring and also an idempotent element in a ring. Definition 1. An element a of a ring R is called idempotent if a2 = a. A ring R is called idempotent if every element of R is idempotent, that is, if a2 = a ∀a ∈ R. M. Shah and T. Shah 2198 In the following we prove that this does not hold in LA-ring. Theorem 2. An LA-ring R cannot be idempotent. Proof. Let a ∈ R then a = a2 = (a + 0)2 = a + 0 by Corollary 6 and by hypothesis, we have (R, +) is abelian group. Hence R cannot be idempotent. The following is another distinctive property of LA-ring. Theorem 3. Let R be an LA-ring R with left identity e. Then right distributive implies left distributive. Proof. Let R be right distributive. Then (b + c)a = ba + ca ∀a, b, c ∈ R. This implies that a(b + c) = (ba + ca)e = (ba)e + (ca)e by right distributivity = ab + ac by left invertive law. Thus R is left distributive. The following theorem presents a beauty of LA-ring. Theorem 4. Let R be an LA-ring R with left identity e then (ab + cd)2 = (ba + dc)2 = (dc + ba)2 ∀a, b, c, d ∈ R. Proof. (ab + cd)2 = = = = = = = (ab + cd)(ab + cd) ab(ab + cd) + cd(ab + cd) by left distributive law [(ab)2 + (ab)(cd)] + [(cd)(ab) + (cd)2 ] by left distributive law [(ba)2 + (dc)(ba)] + [(ba)(dc) + (dc)2 ] by(ab)(cd) = (dc)(ba) (ba + dc)(ba) + (ba + dc)(dc) by right distributive law (ba + dc)(ba + dc) by left distributive law (ba + dc)2 . The second equality follows by Theorem 1. 2199 References [1] J. R. Cho, Pusan, J. Jezek and T. Kepka, Praha, “Paramedial groupoids”, Czechoslovak Mathematical Journal, 49(124)(1996), Praha. [2] P. Holgate, “Groupoids satisfying a simple invertive law”, Math. Stud., 61(1992), 101 − 106. [3] M.S. Kamran, “Conditions for LA-semigroups to resemble associative structures”, Ph.D. Thesis, Quaid-i-Azam University, Islamabad, 1993. Available at http://eprints.hec.gov.pk/2370/1/2225.htm. [4] M. A. Kazim and M. Naseerudin, “On almost semigroups”, Aligarh. Bull. Math., 2(1972)1 − 7. [5] Q. Mushtaq and M.S. Kamran, “On left almost groups”, Proc. Pak. Acad. of Sciences, 33(1996), 1 − 2. [6] Q. Mushtaq and S. M. Yusuf,“On locally associative LA-semigroups”, J. Nat. Sci. Math. Vol. XIX, No. 1, April 1979, pp.57 − 62. [7] M. Shah, A. Ali,“Some structural properties of AG-group”, IMF, Vol. 6, 2011, no. 34, 1661 - 1667. [8] T. Shah and I. Rehman, “ On LA-Rings of finite nonzero functions”, Int. J. Contemp. Math. Sciences, Vol. 5, 2010, no. 5, 209 − 222. [9] S. M. Yusuf, “ On Left Almost Ring”, Proc. of 7th International Pure Math. Conference 2006 (to appear). Received: January, 2011