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On some characterization of Smarandache-boolean near-ring with sub-direct sum structure.

[section]1. Preliminaries

Definition 1.1. A (Left) near ring A is a system with two Binary operations, addition and multiplication, such that

(i) the elements of A form a group (A, +) under addition,

(ii) the elements of A form a multiplicative semi-group,

(iii) x(y + z) = xy + xz, for all x, y and z [member of] A. In particular, if A contains a multiplicative semi-group S whose elements generates (A, +) and satisfy,

(iv) (x + y)s = xs + ys, for all x, y [member of] A and s [member of] S, then we say that A is a distributively generated near-ring.

Definition 1.2. A near-ring (B, +, *) is Boolean-near-ring if there exists a Boolean-ring (A, +, [and], 1) with identity such that is defined in terms of +, [and] and 1, and for any b [member of] B, b * b = b.

Definition 1.3. A near-ring (B, +, *) is said to be idempotent if [x.sup.2] = x, for all x [member of] B. i.e. If (B, +, *) is an idempotent ring, then for all a, b [member of] B, a + a = 0 and a * b = b * a.

Definition 1.4. Compatibility a [member of] b i.e. "a is compatibility to b" if a[b.sup.2] = [a.sup.2]b.

Definition 1.5. Let A = (..., a, b, c, ...) be a set of pairwise compatible elements of an associate ring R. Let A be maximal in the sense that each element of A is compatible with every other element of A and no other such elements may be found in R. Then A is said to be a maximal compatible set or a maximal set.

Definition 1.6. If a sub-direct sum R of domains has an identity, and if R has the property that with each element a, it contains also the associated idempotent [a.sup.0] of a, then R is called an associate subdirect sum or an associate ring.

Definition 1.7. If the maximal set A contains an element u which has the property that a < u, for all a [member of] A, then u is called the uni-element of A.

Definition 1.8. Left zero divisors are right zero divisors, if ab = 0 implies ba = 0.

Now we have introduced a new definition by [3].

Definition 1.9. A Boolean-near-ring (B, [disjunction], [and]) is said to be Samarandache-Boolean-near-ring whose proper subset A is a Boolean-ring with respect to same induced operation of B.

Theorem 1.1. A Boolean-near-ring (B, [disjunction], [and]) is having the proper subset A, is a maximal set with uni-element in an associate ring R, with identity under suitable definitions for (B, +,) with corresponding lattices (A, [less than or equal to])(A, <) and

a [disjunction] b = a + b - 2[a.sup.0] b = (a [union] b) - (a [intersection] b),

a [and] b = a [intersection] b = [a.sup.0]b = a[b.sup.0].

Then B is a Smarandache-Boolean-near-ring.

Proof. Given (B, [disjunction], [and]) is a Boolean-near-ring whose proper subset (A, [disjunction], [and]) is a maximal set with uni-element in an associate ring R, and if the maximal set A is also a subset of B.

Now to prove that B is Smarandache-Boolean-near-ring. It is enough to prove that the proper subset A of B is a Boolean-ring. Let a and b be two constants of A, if a is compatible to b, we define a [and] b as follows:

If [a.sub.i] = [b.sub.i] in the i-component, let [(a [and] b).sub.i] = [0.sub.i];

if [a.sub.i] [not equal to] [b.sub.i], then since a ~ b precisely one of these is zero;

if [a.sub.i] = 0, let [(a [and] b).sub.i] = [b.sub.i] [not equal to] 0;

if [b.sub.i] = 0, let [(a [and] b).sub.i] = [a.sub.i] [not equal to] 0.

It is seen that if a [and] b belongs to the associate ring R, then a [and] b < u, where u is the uni-element of A, and therefore, a [and] b [member of] A.

Consider a [and] b = a + b - 2[a.sup.0]b:

If in the i-component, 0 [not equal to] [a.sub.i]-[b.sub.i], then since [([a.sup.0]).sub.i] = [1.sub.i] = [([b.sup.0]).sub.i], we have [(a + b - 2[a.sup.0]b).sub.i] = [0.sub.i];

if [0.sub.i] = [a.sub.i] = [b.sub.i], then [([a.sup.0]).sub.i] = 0 and [([b.sup.0]).sub.i] = 1, whence, [(a + b- 2[a.sup.0]b).sub.i] = [b.sub.i];

if [a.sub.i] [not equal to] 0 and [b.sub.i] = 0 then (a + b - 2[a.sup.0]b) = [0.sub.i].

Therefore a [and] b [member of] A, the maximal set.

Similarly, the element a [and] b = a [intersection] b = [a.sup.0]b = a[b.sup.0] = glb(a, b) has defined and shown to belongs to A as the glb(a, b). Now let us show that (A, [disjunction], [and]) is a Boolean-ring. Firstly, a [disjunction] a = 0, since [a.sub.i] = [a.sub.i] in every i-component, whence [(a[disjunction]a).sub.i] vanishes, by our definition of '[and]'. Secondly a [and] a = a [intersection] a = [a.sup.0]a = a, and so a is idempotent under [and]. We have shown that A is closed under [and] is [disjunction], and associativity is a direct verification, and each element is itself inverse under [and].

To prove associativity under [and]:

For [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For distributivity of [and] over [and], let c be an arbitrary element in A.

Now [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Hence (A, [disjunction], [and]) is a Boolean-ring.

It follows that the proper subset A, a maximal set of B forms a Boolean ring. B is a Boolean-near-ring, whose proper subset is a Boolean-ring, then by definition, B is a SmarandacheBoolean-near-ring.

Theorem 1.2. A Boolean-near-ring (B, [disjunction], [and]) is having the proper subset (A, +, [and], 1) is an associate ring in which the relation of compatibility is transitive for non-zero elements with identity under suitable definitions for (B, +, *) with corresponding lattices (A, [less than or equal to])(A, <) and

a [disjunction] b = a + b - 2[a.sup.0]b = (a [union] b) - (a [intersection] b),

a [and] b = a [intersection] b = [a.sup.0]b = a[b.sup.0].

Then B is a Smarandache-Boolean-near-ring.

Proof. Assume that (B, +, *) be Boolean- near-ring having a proper subset A is an associate ring in which the relation of compatibility is transitive for non-zero elements.

Now to prove that B is a Smarandache-Boolean-near-ring, i.e., to prove that if the proper subset of B is a Boolean-ring, then by definition B is Smarandache-Boolean-near-ring. We have 0 is compatible with all elements, whence all elements are compatible with A and therefore, are idempotent.

Then assume that transitivity holds for compatibility of non-zero elements. It follows that non-zero elements from two maximal sets cannot be compatible (much less equal), and hence, except for the element 0, the maximal sets are disjoint.

Let a be a arbitrary, non-zero element of R. If a is a zero-divisor of R, then the idempotent element A - [a.sup.0] [not equal to] 0. Further A - [a.sup.0] belongs to the maximal set generated by the non-zero divisor a' = a + A - [a.sup.0], since it is (A-[a.sup.0])a' = (A - [a.sup.0])(a + A-[a.sup.0]) = (A - [a.sup.0]) = [(A - [a.sup.0]).sup.2] i.e. A - [a.sup.0] < a'. Since also a < a' and a ~ A - [a.sup.0], therefore, a is idempotent. i.e. All the zero-divisors of R are idempotent which is a maximal set then by theorem 1 and by definition A is a Boolean-ring. Then by definition, B is Smarandache-Boolean-near-ring.

Theorem 1.3. A Boolean-near-ring (B, [disjunction], [and]) is having the proper subset A, the set A of idempotent elements of a ring R, with suitable definitions for [disjunction] and [and],

a [disjunction] b = a + b - 2[a.sup.0]b = (a [union] b) - (a [intersection] b),

a [and] b = a [intersection] b = [a.sup.0]b = a[b.sup.0].

Then B is a Smarandache-Boolean-near-ring.

Proof.

Assume that the set A of idempotent elements of a ring R, which is also a subset of B. Now to prove that B is a Smarandache-Boolean-near-ring. It is sufficient to prove that the set A of idempotent elements of a ring R with identity forms a maximal set in R with uni-element. By the definition of compatible, then we have every element of R is compatible with every other idempotent element. If a [member of] R is not idempotent then, [a.sup.2] * 1 [not equal to] a * [1.sup.2], since the definition of compatible. Hence no non-idempotent can belong to this maximal set. Thus the set A is idempotent element of R with identity forms a maximal set in R whose uni-element is the identity of R, by theorem 1 and by definition. A, a maximal set of B forms a Boolean ring

Then by definition, it concludes that B is Smarandache-Boolean-near-ring.

Theorem 1.4. A Boolean-near-ring (B, [disjunction], [and]) is having the proper subset, having a nonzero divisor of A, as an associate ring, with suitable definitions for [disjunction] and [and],

a [disjunction] b = a + b - 2[a.sup.0]b = (a [union] b) - (a [intersection] b)

a [and] b = a [intersection] b = [a.sup.0]b = a[b.sup.0].

Then B is a Smarandache-Boolean-near-ring.

Proof. Let B is Boolean-near-ring whose proper subset having a non-zero divisor of associate ring A.

Now to prove that B Smarandache-Boolean-near-ring. It is enough to prove that every non-divisor of A determines uniquely a maximal set of A with uni-element.

Let a be the uni-element of a maximal set A then we have b < a, for b [member of] A.

Consider all the elements of A in whose sub-direct display one or more component [a.sub.i] duplicate the corresponding component [u.sub.i] of u, the other components of a being zeros, i.e., all the element a such that a < u, becomes u is uni-element. Clearly, these elements are compatible with each other and together with u form a maximal set in A, for which u is the uni-element. Hence A is a maximal set with uni-element and by theorem 1 and definition A, a maximal set of B forms a Boolean ring.

Then by definition, B is Smarandache-Boolean-near-ring.

Theorem 1.5. A Boolean-near-ring (B, [disjunction], [and]) is having the proper subset A, associate ring is of the form A = [u.sub.J], where u is a non-zero of A and J is the set of idempotent elements of A, with suitable definitions for [disjunction] and [and],

a [disjunction] b = a + b - 2[a.sup.0]b = (a [union] b) - (a [intersection] b),

a [and] b = a [intersection] b = [a.sup.0]b = a[b.sup.0].

Then B is a Smarandache-Boolean-near-ring.

Proof. Assume that the proper subset A of a Boolean-near-ring B is of the form A = [u.sub.J], where u is non-zero divisor of A and J is the set of idempotent elements of A. Now to prove B is Smarandache-Boolean-near-ring. It is enough to prove that A is a maximal set with uni-element.

(i) It is sufficient to show that the set [u.sub.J] is a maximal set having u as its uni-element.

(ii) If b belongs to the maximal set determined by u, then b has the required form b = [u.sub.e], for some e [member of] J.

Proof of (i). It is seen that [u.sub.e] ~ [u.sub.f] i.e. [u.sub.e] is compatible to [u.sub.f] with uni-element u, for all e, f [member of] J, since idempotent belongs to the center of A. Also, [u.sub.e] < u, since [u.sub.e] * u = [u.sup.2.sub.e] = [([u.sub.e]).sup.2].

Proof of (ii). We know that A is an associate ring, the associated idempotent [a.sup.0] of a has the property: if a ~ b then [a.sup.0]b = a[b.sup.0] = [b.sup.0]a = b[a.sup.0]; if a [member of] [A.sub.u], then since a < u and [u.sup.0] = 1, we have A = [u.sup.0]a = a[u.sup.0] = [a.sup.0]u, for all [a.sup.0] [member of] J.

Hence A is a maximal set with uni-element of of B by suitable definition and by theorem 1 then we have A is a Boolean-ring. Then by definition, B is Smarandache-Boolean-near-ring.

References

[1] G. Bermann and R. J. Silverman, Near Rings, Amer. Math., 66(1954).

[2] Birkhoff, Garrett, Lattice Theory, American. Math. Soc, Colloquium Publication, 1939.

[3] Florentin Smarandache, Special algebraic structures, University of Maxico, Craiova, 1973.

[4] A. L. Foster, Generalized boolean theory of universal algebras, Math. Z., 1958.

[5] A. L. Foster, The idempotent elements of a commutative ring form a Boolean Algebra, Duke math. J., 1946.

[6] James R. Clay and Donald A. Lawver, Boolean-near-rings, Canad. Math. Bull., 1968.

[7] N. H. McCoy and D. Montgomery, A representation of generalized Boolean rings, Duke Math. J., 1937.

[8] N. H. McCoy, Subdirect sums of rings, Bull. Amer. Math. Soc, 1947.

[9] Padilla Raul, Smarandache Algebraic Structures, Delhi, India, 1998.

[10] Padilla Raul, Smarandache Algebraic Structures, USA, 9(1998).

[11] G. Pilz, Near rings, North Holland Press, Amsterdam, 1977.

[12] T. Ramaraj and N. Kannappa, On finite Smarandache-near-rings, Scientia Magna, 1(2005), No. 2, 49-51.

Dr. N. Kannappa ([dagger]) and Mrs. P. Tamilvani ([double dagger])

([dagger]) TBML College, Porayar-609307, Tamilnadu, India

E-mails: sivaguru91@yahoo.com
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Author:Kannappa, N.; Tamilvani, P.
Publication:Scientia Magna
Article Type:Technical report
Date:Sep 1, 2013
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