Download Some Basic Properties of LA-Ring

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
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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
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3
3
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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
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4
2
0
2
4
1
3
3
0
3
1
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0
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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
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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.
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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.
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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