Direct Sum of Ideals in a Generalized LA-Ring

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1 International Mathematical Forum, Vol. 6, 2011, no. 22, Direct Sum of Ideals in a Generalized LA-Ring Tariq Shah, Gauhar Ali and Fazal ur Rehman Department of Mathematics Quaid-i-Azam University, Islamabad, Pakistan stariqshah@gmail.com, gauharali4@gmail.com Abstract. A near LA-ring (nla-ring) is a generalization of an LA-ring and footed parallel to the near ring. In this study we characterized nla-ring through its ideals. We have shown that the sum of ideals is again an ideal, and established the necessary and sufficient condition for an nla-ring to be direct sum of its ideals. Furthermore we observed that the product of ideals is just a left ideal. Mathematics Subject Classification: 16A76, 20M25, 20N02 Keywords: LA-ring, nla-ring, ideals 1. Introduction and Preliminaries The concept of left almost semigroup is introduced by M. Kazim and M. Naseeruddin in [1], which generalizes the notion of a commutative semigroup. A groupoid S is called a left almost semigroup (abbreviated as an LA-semigroup) if its elements satisfy the left invertive law: (ab) c =(cb) a for all a, b, c S. It is also called an Abel-Grassmann s groupoid (abbreviated as an AG-groupoid), for instance see [7]. Later, the structure was explored in [5] by Q. Mushtaq and S. M. Yusuf. Q. Mushtaq and Madad Khan proceed the structure of an LA-semigroup through its ideals in [3, 4]. Further in [2] Q. Mushtaq and S. Kamran extend the concept to left almost groups. A left almost group G is a non-empty set with one binary operation such that (G, ) is an LAsemigroup, having left identity e and each element of G has left inverse. The left almost group though is a non abelian structure has interesting resemblance with a commutative group. S. M. Yusuf in [10] introduces the concept of a left almost ring. That is, a non-empty set R with two binary operations + and is called a left almost ring, if (R, +) is an LA-group, (R, ) is an LA-Semigroup and distributive

2 1096 Tariq Shah, Gauhar Ali and Fazal ur Rehman law of over + holds. Further in [9] Tariq Shah and Inayatur-Rehman generalize the notions of commutative semigroup rings into LA-semigroup LArings. However Tariq Shah and Fazal ur Rehman in [9] generalize the notion of an LA-ring into an nla-ring. A near left almost ring (nla-ring) N is an LA-group under +, an LA-semigroup under and left distributive property of over + holds. An nla-subring I of an nla-ring N is a left ideal if NI I and is called a right ideal if for all i I and n, m N such that (i + n) m nm I. In this paper, we characterize the structure of near LA-ring through its ideals. We discuss sum, direct sum and product of ideals in an nla-ring. 2. Direct Sum of ideals In this section, we discuss sum and direct sum of ideals in an nla-ring. Definition 1. [9] A non empty set N with two binary operations + and is called a near left almost ring (or simply nla-ring) if and only if (nla1) (N,+) is an LA-group. (nla2) (N, ) is an LA-semigroup. (nla3) Left distributive property of over + holds. Definition 2. An nla-subring I of an nla-ring N is called a left ideal of N if NI I, and I is called a right ideal if for all n, m N and i I such that (i + n) m nm I. And is called two sided ideal or simply ideal if it is both left and right ideal. Proposition 1. Let I and J be two ideals in nla-ring N. Then I + J =< I J>is an ideal. Proof. Since 0 = I + J, so I + J φ and obviously I + J is a left ideal of N. Let x = i + j I + J, where i I, j J and n 1,n 2 N. Consider (x + n 1 ) n 2 n 1 n 2 = ((i + j)+n 1 ) n 2 n 1 n 2 =((i +0)+(j + n 1 )) n 2 n 1 n 2 = ( n 1 n 2 +((i +0)+(j + n 1 )) n 2 )+0 = ( n 1 n 2 +((i +0)+(j + n 1 )) n 2 )+((j + n 1 )n 2 (j + n 1 )n 2 ) = ( n 1 n 2 +(j + n 1 )n 2 ) + (((i +0)+(j + n 1 ))n 2 (j + n 1 )n 2 ) = (((j + n 1 )n 2 n 1 n 2 ) + 0) + (((i +0)+(j + n 1 ))n 2 (j + n 1 )n 2 ) = ((((i +0)+(j + n 1 ))n 2 (j + n 1 )n 2 ) + 0) + ((j + n 1 )n 2 n 1 n 2 ). This means (x + n 1 ) n 2 n 1 n 2 I + J and hence I + J is an ideal of N. As I = I +0 I + J and J =0+J I + J, soi J I + J. Let L be any ideal of N such that I J L and x = i + j I + J, where i I and j J. Obviously i, j I J L and x = i + j L. This implies that I + J L and hence I + J =< I J>.

3 Direct sum of ideals in a generalized LA-ring 1097 Since an nla-ring is a non-associative and non-commutative structure, therefore a question arises that in which form the elements of sum of more than two ideals appear. This problem is overcome by defining the sum as follows: Definition 3. Let (I k ) k K be a family of ideals of an nla-ring N. The sum of ideals (I k ) k K is defined as having the representation I k = (... ((I 1 + I 2 )+I 3 )+... k K = { } ik =(... ((i 1 + i 2 )+i 3 )+... : i k I k, where {I i : i =1, 2, 3...} are ideals of nla-ring N. Proposition 2. If N is an nla-ring and F = {I n : n Λ} is a family of left ideals of N, then (F, +) is an LA-semigroup. Where + is the sum of ideals defined in the Definition 3. Lemma 1. If (S, ) is an LA-monoid with left identity e, then (a b) c = (a (c e)) (b e) holds for all a, b, c S. The following generalizes the Lemma 1. Lemma 2. If (S, ) is an LA-monoid with left identity e, then position of any element a i in the product of elements of S can be changed accordingly. (1) If n i is even, then ((... ((a 1 a 2 ) a 3 )...) a i )...) a n 1 ) a n = ((... ((a 1 a 2 ) a 3 )...) a i 1 ) (a i+1 e))...) (a n 1 e)) (a n e)) a i. (2) If n i is odd, then ((... ((a 1 a 2 ) a 3 )...) a i )...) a n 1 ) a n = ((... ((a 1 a 2 ) a 3 )...) a i 1 ) (a i+1 e))...) (a n 1 e)) (a n e)) (a i e), where a i S, i =1, 2,..., n. Proof. (1) Let n i = k, Take x =((... (a 1 a 2 ) a 3...) a i 1 ). By mathematical induction let k =2, i.e. ((x a i ) b 1 ) b 2 = ((x (b 1 e)) (a i e)) b 2 = ((x (b 1 e)) (b 2 e)) a i. This shows that it is true for k =2. Suppose that the statement is true for k =2t, where 1 <t Z +, i.e. (... ((x a i ) b 1 ) b 2 )...) b 2t 1 ) b 2t = ((... (x (b 1 e)) (b 2 e))...) (b 2t 1 e)) (b 2t e)) a i.

4 1098 Tariq Shah, Gauhar Ali and Fazal ur Rehman Now we prove that the statement is true for k =2t +2. Consider (... ((x a i ) b 1 ) b 2 )...) b 2t 1 ) b 2t ) b 2t+1 ) b 2t+2 = ((... (x (b 1 e)) (b 2 e))...) (b 2t 1 e)) (b 2t e)) a i ) b 2t+1 ) b 2t+2 = ((... (x (b 1 e)) (b 2 e))...) (b 2t 1 e)) (b 2t e)) (b 2t+1 e)) (a i e)) b 2t+2 = ((... (x (b 1 e)) (b 2 e))...) (b 2t 1 e)) (b 2t e)) (b 2t+1 e)) (b 2t+2 e)) a i. (2) Same as (1). The converse of lemma 2 is given below. Lemma 3. If (S, ) is an LA-monoid with left identity e, then position of any element a i in the product of elements of S can be changed accordingly. (1) If n i is even, then (((... ((a 1 a 2 ) a 3 )...) a i 1 ) a i+1 )...) a n 1 ) a n ) a i = ((... ((a 1 a 2 ) a 3 )...) a i 1 ) a i ) (a i+1 e))...) (a n 1 e)) (a n e) (2) If n i is odd, then (((... ((a 1 a 2 ) a 3 )...) a i 1 ) a i+1 )...) a n 1 ) a n ) a i = ((... ((a 1 a 2 ) a 3 )...) a i 1 ) (a i + e)) (a i+1 e))...) (a n 1 e)) (a n e), where a i S, i =1, 2,..., n. Lemma 4. Let (S, ) be an LA-monoid with left identity e. If a e = e, then a = e for all a S. Definition 4. Let {I k : k =1, 2,..., n} be a family of ideals of an nla-ring N. Their sum n I k =((... ((I 1 + I 2 )+I 3 )...)+I n 1 )+I n is called an internal n direct sum if each element of has a unique representation as a I k sum of different I k s. In this case, we write for the direct sum as ((... ((I 1 I 2 ) I 3 )...) I n 1 ) I n. n I k = Theorem 1. Necessary and sufficient condition for an nla-ring N to be the direct sum of {I k : k =1, 2,..., n} if (1) N = n I k. (2) n I k I i = {0}.

5 Direct sum of ideals in a generalized LA-ring 1099 Proof. Necessary condition. Suppose that for each 0 m N can be uniquely express as m =((... ((i 1 + i 2 )+i 3 )...)+i n 1 )+ i n then clearly N = n I k is true. Now for (2) let x n I k I i = x n I k and x I i then x = ((... ((i 1 + i 2 )+i 3 )...)+i i 1 )+i i+1 )...)+i n 1 )+i n and x = i i = i i =((... ((i 1 + i 2 )+i 3 )...)+i i 1 )+i i+1 )...)+i n 1 )+i n = 0 = (((... ((i 1 + i 2 )+i 3 )...)+i i 1 )+i i+1 )...)+i n 1 )+i n ) i i.(1) Here we have two cases: Case 1: If n i is even, then from equation (1). 0 = ((... ((i 1 + i 2 )+i 3 )...)+i i 1 ) i i )+(i i+1 + 0))...)+(i n 1 + 0)) +(i n +0), by Lemma 3. = ((... ((0 + 0) + 0)...) + 0) + 0) + 0)...)+0)+0 = ((... ((i 1 + i 2 )+i 3 )...)+i i 1 ) i i )+(i i+1 + 0))...)+(i n 1 + 0)) + (i n +0) = i k = 0 for all k =1, 2, 3,..., n, by Lemma 4. Case 2: If n i is odd, then from equation (1) and by Lemma 3, we have 0 = ((... ((i 1 + i 2 )+i 3 )...)+i i 1 )+( i i +0))+(i i+1 + 0))...)+(i n 1 + 0)) + (i n +0) This implies ((... ((0 + 0) + 0)...) + 0) + 0) + 0)...)+0)+0 = ((... ((i 1 + i 2 )+i 3 )...)+i i 1 )+( i i +0))+(i i+1 + 0))...)+(i n 1 + 0)) + (i n +0). This implies i k = 0 for all k =1, 2, 3,..., n, by Lemma 4. Hence n I k I i = {0}. For sufficient condition, let 0 n N be such that n =(... ((i 1 + i 2 )+i 3 )...)+i i )...)+i n 1 )+i n and n =(... ((j 1 + j 2 )+j 3 )...)+j i )...)+j n 1 )+j n, where i k,j k I k, now (... ((i 1 + i 2 )+i 3 )...)+i i )...)+i n 1 )+i n = (... ((j 1 + j 2 )+j 3 )...)+j i )...)+j n 1 )+j n. Therefore 0 = ((... ((j 1 + j 2 )+j 3 )...)+j i )...)+j n 1 )+j n ) ((... ((i 1 + i 2 )+i 3 )...)+i i )...)+i n 1 )+i n ). 0 = ((... (((j 1 i 1 )+(j 2 i 2 ))+(j 3 i 3 ))...)+(j i i i ))...) +(j n 1 i n 1 ))+(j n i n ). (2) Here we have two cases;

6 1100 Tariq Shah, Gauhar Ali and Fazal ur Rehman Case 1 :Ifn iis even, then from equation (2). 0 = (... (((j 1 i 1 )+(j 2 i 2 ))+(j 3 i 3 ))...)+(j i 1 i i 1 )) + (j i 1 i i 1 )+0)...)+((j n 1 i n 1 )+0))+((j n i n ) + 0)) + (j i i i ) by Lemma 2 (j i i i ) = (... (((j 1 i 1 )+(j 2 i 2 ))+(j 3 i 3 ))...)+(j i 1 i i 1 )) + ((j i 1 i i 1 ) + 0))...)+((j n 1 i n 1 ) + 0)) + ((j n i n ) + 0)) implies that (j i i i ) n I k I i = {0} Implies that (j i i i )=0 = j i = i i. Case 2 :Ifn i is odd, then from equation 2. 0 = (... (((j 1 i 1 )+(j 2 i 2 ))+(j 3 i 3 ))...)+(j i 1 i i 1 )) + ((j i 1 i i 1 ) + 0))...)+((j n 1 i n 1 ) + 0)) + ((j n i n ) + 0)) +((j i i i )+0), by Lemma 2. ((j i i i )+0) = (... (((j 1 i 1 )+(j 2 i 2 ))+(j 3 i 3 ))...)+(j i 1 i i 1 )) + ((j i 1 i i 1 ) + 0))...)+((j n 1 i n 1 ) + 0)) + ((j n i n ) + 0)). This implies that ((j i i i )+0) n I k I i = {0} and so ((j i i i ) + 0) = 0 and hence j i = i i. 3. Product of ideals Definition 5. Let I and J be two ideals of an nla-ring N. Then their product is defined by IJ = { a ib i =(... ((a 1 b 1 + a 2 b 2 )+a 3 b 3 ) a n 1 b n 1 )+a n b n ; where a i I and b i J }. Lemma 5. If (S, ) is an LA-monoid with left identiity e, then (a b) (c d) = ((b a) c) d for all a, b, c, d S. Proposition 3. Let I and J be two ideals of an nla-ring N. Then IJ is a left ideal of N contained in I J. Proof. Let x, y IJ, then x = a i b i and y = c i d i, where a i,c i I and b i,d i J. Now by using medial law and lemma 5, we have x y = a i b i c i d i = x i y i IJ, where x i y i {0,a i b i,c j d j }. i

7 Direct sum of ideals in a generalized LA-ring 1101 Let n N, then ( ) nx = n a i b i = n (a i b i )= a i (nb i ) IJ. Hence IJ is a left ideal of N. Let for any x = a ib i IJ. Since b i J and J is ideal so a i b i J and therefore x J. Also a i b i = (a i b i 0) + 0 = (((a i +0)+0)b i 0) + 0 = (((a i +0)+0)b i 0b i )+0 I, as I is an ideal. This means x I. Hence IJ I J. References [1] M.A. Kazim and M. Naseeruddin, On almost semigroups, The Alig. Bull. Math. 2 (1972) 1-7. [2] Q. Mushtaq and M. S. Kamran, Left almost group, Proc. Pak. Acad. of Sciences. 33 (1996) 1-2. [3] Q. Mushtaq and M. Khan, Ideals in left almost semigroups, Proc. of 3rd International Pure Math. Conference [4] Q. Mushtaq and M. Khan, Direct Product of Abel Grassmann s groupoids, Journal of interdisciplinary Mathematics Vol. 11 (2008), [5] Q. Mushtaq and S. M. Yusuf, On LA-semigroups, The Alig. Bull. Math. 8 (1978) [6] G. Pilz, Near-rings, North Holland, Amesterdam-New York, (1977) [7] P.V. Protic, N. Stevanovi, AG-test and some general properties of Abel- Grassmann s groupoid, P.U.M.A. Vol. 6 (1995), No. 4, pp [8] T. Shah and I. Rehman, On LA-Rings of Finitely Nonzero Functions, Int. J. Contemp. Math. Sciences, Vol. 5 (2010), no. 5, [9] T. Shah, Fazal ur Rehman and M. Raees, On Near Left Almost Rings, (to appear). [10] S.M. Yusuf, On Left Almost Ring, Proc. of 7th International Pure Math. Conference (2006). Received: April, 2010