# Smarandache quasigroup rings.

Abstract In this paper, we have introduced Smarandache quasigroups
which are Smarandache non-associative structures. W.B.Kandasamy [2] has
studied groupoid ring and loop ring. We have defined Smarandache
quasigroup rings which are again non-associative structures having two
binary operations. Substructures of quasigroup rings are also studied.

Keywords Non-associative rings; Smarandache non-associative rings; Quasigroups; Smarandache quasigroups; Smarandache quasigroup rings.

[section] 1. Introduction

In the paper [2] W.B.Kandasamy has introduced a new concept of groupoid rings. This structure provides number of examples of SNA-rings (Smarandache non-associative rings). SNA-rings are non-associative structure on which are defined two binary operations one associative and other being non-associative and addition distributes over multiplication both from right and left. We are introducing a new concept of quasigroup rings. These are non associative structures. In our view groupoid rings and quasigroup rings are the rich source of non-associative SNA-rings without unit since all other rings happen to be either associative or non-associative rings with unit. To make this paper self contained we recollect some definitions and results which we will use subsequently.

[section] 2. Preliminaries

Definition 2.1. A groupoid S such that for all a, b [member of] S there exist unique x, y [member of] S such that ax = b and ya = b is called a quasigroup.

Thus a quasigroup does not have an identity element and it is also non-associative. Here is a quasigroup that is not a loop.

[ILLUSTRATION OMITTED]

We note that the definition of quasigroup Q forces it to have a property that every element of Q appears exactly once in every row and column of its operation table. Such a table is called a LATIN SQUARE. Thus, quasigroup is precisely a groupoid whose multiplication table is a LATIN SQUARE.

Definition 2.2. If a quasigroup (Q, *) contains a group (G, *) properly then the quasigroup is said to be Smarandache quasigroup. Example 2.1. Let Q be a quasigroup defined by the following table:

[ILLUSTRATION OMITTED]

Clearly, A = {[a.sub.0], [a.sub.1]} is a group w.r.t. * which is a proper subset of Q. Therefore Q is a Smarandache quasigroup.

Definition 2.3. A quasigroup Q is idempotent if every element x in Q satisfies x * x = x.

Definition 2.4. A ring (R, +, *) is said to be a non-associative ring if (R, +) is an additive abelian group, (R, *) is a non-associative semigroup (i.e. binary operation * is non-associative)such that the distributive laws a * (b + c) = a * b + a * c and (a + b) * c = a * c + b * c for all a, b, c in R.

Definition 2.5. Let R be a commutative ring with one. G be any group (S any semigroup with unit) RG (RS the semigroup ring of the semigroup S over the ring R) the group ring of the group G over the ring R consists of finite formal sums of the form [n.summation over (i=1)] [[alpha].sub.i][g.sub.i], (n < [infinity]) i.e. i runs over a finite number where [[alpha].sub.i] [member of] R and [g.sub.i] [member of] G ([g.sub.i] [member of] S) satisfying the following conditions:

1. [n.summation over (i=1)] [[alpha].sub.i][m.sub.i] = [n.summation over (i=1)] [[beta].sub.i][m.sub.i] [left and right arrow] [[alpha].sub.i] = [[beta].sub.i], for i = 1, 2,..., n

2. [n.summation over (i=1)] [[alpha].sub.i][m.sub.i] + [n.summation over (i=1)] [[beta].sub.i][m.sub.i] [left and right arrow] [n.summation over (i=1)] ([[alpha].sub.i] + [[beta].sub.i])[m.sub.i]

3. ([n.summation over (i=1)] [[alpha].sub.i][m.sub.i])([n.summation over (i=1)] [[beta].sub.i][m.sub.i]) = [n.summation over (i=1)] [[gamma].sub.k][m.sub.k],[m.sub.k] = [m.sub.i][m.sub.j], where [[gamma].sub.k] = [summation][[alpha].sub.i][[beta].sub.i]

4. [r.sub.i][m.sub.i] = [m.sub.i][r.sub.i] for all [r.sub.i] [member of] R and [m.sub.i] [member of] G([m.sub.i] [membr fo] S).

5. r [n.summation over (i=1)] [r.sub.i][m.sub.i] = [n.summation over (i=1)] r[r.sub.i][m.sub.i] for all r [member of] R and [n.summation over (i=1)] [r.sub.i][m.sub.i] [member of] RG. RG is an associative ring with 0 [member of] R acts as its additive identity. Since I [member of] R we have G = IG [[subset].bar] RG and R:e = R [[subset].bar] RG where e is the identity element of G.

If we replace the group G in the above definition by a quasigroup Q we get RQ the quasigroup ring which will satisfy all the five conditions 1 to 5 given in the definition. But RQ will only be a non-associative ring without identy. As I [member of] R we have Q [[subset].bar] RQ. Thus we define quasigroup rings as follows:

Definition 2.6. For any quasigroup Q the quasigroup ring RQ is the quasigroup Q over the ring R consisting of all finite formal sums of the form [n.summation over (i=1)] [r.sub.i][q.sub.i], (n < [infinity]) i.e. i runs over a finite number where [r.sub.i] [member of] R and [q.sub.i] [member of] Q satisfying conditions 1 to 5 given in the definition of group rings above.

Note that only when Q is a quasigroup with identity (i.e. then Q is a Loop) that the quasigroup ring RQ will be a non-associative ring with unit. Here we give examples of non-associative quasigroup rings.

Example 2.2. Let Z be the ring of integers and (Q, *) be the quasigroup given by the following table:

[ILLUSTRATION OMITTED]

Clearly (Q, *) is a quasigroup and does not posses an identity element. The quasigroup ring ZQ is a non-associative ring without unit element.

Example 2.3. Let R be the ring of reals and (Q, *) be the quasigroup defined by the following table:

[ILLUSTRATION OMITTED]

(Q, *) is an idempotent quasigroup. Again RQ is a non-associative quasigroup ring without unit. Note that R<1>, R<2>, R<3>, R<4> are the subrings of RQ which are associative.

Result: All quasigroup rings RQ of a quasigroup Q over the ring R are non-associative rings without unit.

The smallest non-associative ring without unit is quasigroup ring given by the following example. This example was quoted by W.B.Kandasamy [2] as a groupoid ring.

Example 2.4. Let [Z.sub.2] = {0, 1} be the prime field of characteristic 2. (Q, *) be a quasigroup of order 3 given by the following table:

[ILLUSTRATION OMITTED]

[Z.sub.2]Q is a quasigroup ring having only eight elements given by {0, [q.sub.1], [q.sub.2], [q.sub.3], [q.sub.1] + [q.sub.2], [q.sub.2] + [q.sub.3], [q.sub.1] + [q.sub.3], [q.sub.1] + [q.sub.2] + [q.sub.3]}. Clearly, [Z.sub.2]Q is a non-associative ring without unit. This happens to be the smallest non-associative ring without unit known to us.

[section] 3. SNA-Quasigroup rings

We introduce Smarandache non-associative quasigroup rings. It is true that quasigroup rings are always non-associative. We write "Smarandache non-associative quasigroup ring" only to emphasize the fact that they are non-associative.

Definition 3.1. Let S be a quasigroup ring. S is said to be SNA-quasigroup ring (Smarandache non-associative quasigroup ring) if S contains a proper subset P such that P is an associative ring under the operations of S.

Example 3.1. Let Z be the ring of integers and Q be a quasigroup defined by the following table;

[ILLUSTRATION OMITTED]

Clearly, A = {[a.sub.0], [a.sub.1]} is group and ZQ [contains] ZA. Thus the quasigroup ring ZQ contains an associative ring properly. Hence ZQ is an SNA-quasigroup ring. Note that Q is a Smarandache quasigroup.

Example 3.2. Let R be the reals, (Q, *) be the quasigroup defined by the following table;

[ILLUSTRATION OMITTED]

Then clearly RQ is an SNA-quasigroup ring as RQ [contains] R<0, 1> and R<0, 1> is an associative ring.

Theorem 3.1. Let Q be a quasigroup and R be any ring. Then the quasigroup ring RQ is not always an SNA-quasigroup ring.

Proof. Since Q does not have an identity element, there is no guarantee that R is contained in RQ .

Example 3.3. Let R be an arbitrary ring and Q be a quasigroup defined by the table;

[ILLUSTRATION OMITTED]

Then clearly, RQ is not an SNA-quasigroup ring as the quasigroup ring RQ does not contain an associative ring.

Theorem 3.2. If Q is a quasigroup with identity, then quasigroup ring RQ is SNA-quasigroup ring.

Proof. Quasigroup with identity is a Loop. So, RI [[subset].bar] RQ and R serves as the associative ring in RQ. Thus RQ is an SNA-quasigroup ring.

Theorem 3.3. Let R be a ring. If Q is a Smarandache quasigroup, then quasigroup ring RQ is an SNA-quasigroup ring. .

Proof. Obviously RQ is a non-associative ring. As Q is a Smarandache quasigroup Q contains a group G properly. So RQ [contains] RG and RG is an associative ring contained in RQ. Therefore RQ is an SNA-quasigroup ring.

[section] 4. Substructure of SNA-quasigroup rings

Definition 4.1. Let R be a SNA-quasigroup ring. Let S be a non-empty subset of R. Then S is said to be S-quasigroup subring of R if S itself is a ring and contains a proper subset P such that P is an associative ring under the operation of R.

Example 4.1. Let Z be the ring of integers. Let Q be the quasigroup defined by the following table:

[ILLUSTRATION OMITTED]

Clearly the quasigroup ring ZQ is a non-associative ring. Consider the subset S = {1, 2, 3, 4} then S is a group and hence ZS is a group ring and hence also a quasigroup ring. Let P = {1, 2}. Note that ZS also contains ZP where P = {1, 2}. So, ZS is an S-quasigroup subring of SNA-quasigroup ring ZQ.

We have not yet been able to find a Smarandache non associative quasigroup subring for a given quasigroup ring. We think that it is not possible to obtain a subquasigroup for any quasigroup because for a quasigroup its composition table is a LATIN SQUARE.

Theorem 4.1. Let R be a quasigroup ring, if R has a SNA-quasigroup subring S, then R itself is SNA-quasigroup ring.

Proof. As S is an SNA-quasigroup surbring S contains an associative ring. As a result R contains an associtive ring. Thus R is an SNA-quasigroup ring.

References

[1] R.H.Bruck, A survey of binary systems, Springer Verlag, 1958.

[2] W.B.Kandasamy, Smarandache non-associative (SNA) rings, Smarandache Notions (book series), American Research Press, 14(2004), 281-293.

[3] D.S. Passman, The algebraic structure of group rings, Wiley- interscience, 1977.

[4] J.S.Robinson Derek, A course in the theory of Groups, Springer Verlag, 1996. Scientia Magna

Arun S. Muktibodh

Mohota Science College

Nagpur, India

Keywords Non-associative rings; Smarandache non-associative rings; Quasigroups; Smarandache quasigroups; Smarandache quasigroup rings.

[section] 1. Introduction

In the paper [2] W.B.Kandasamy has introduced a new concept of groupoid rings. This structure provides number of examples of SNA-rings (Smarandache non-associative rings). SNA-rings are non-associative structure on which are defined two binary operations one associative and other being non-associative and addition distributes over multiplication both from right and left. We are introducing a new concept of quasigroup rings. These are non associative structures. In our view groupoid rings and quasigroup rings are the rich source of non-associative SNA-rings without unit since all other rings happen to be either associative or non-associative rings with unit. To make this paper self contained we recollect some definitions and results which we will use subsequently.

[section] 2. Preliminaries

Definition 2.1. A groupoid S such that for all a, b [member of] S there exist unique x, y [member of] S such that ax = b and ya = b is called a quasigroup.

Thus a quasigroup does not have an identity element and it is also non-associative. Here is a quasigroup that is not a loop.

[ILLUSTRATION OMITTED]

We note that the definition of quasigroup Q forces it to have a property that every element of Q appears exactly once in every row and column of its operation table. Such a table is called a LATIN SQUARE. Thus, quasigroup is precisely a groupoid whose multiplication table is a LATIN SQUARE.

Definition 2.2. If a quasigroup (Q, *) contains a group (G, *) properly then the quasigroup is said to be Smarandache quasigroup. Example 2.1. Let Q be a quasigroup defined by the following table:

[ILLUSTRATION OMITTED]

Clearly, A = {[a.sub.0], [a.sub.1]} is a group w.r.t. * which is a proper subset of Q. Therefore Q is a Smarandache quasigroup.

Definition 2.3. A quasigroup Q is idempotent if every element x in Q satisfies x * x = x.

Definition 2.4. A ring (R, +, *) is said to be a non-associative ring if (R, +) is an additive abelian group, (R, *) is a non-associative semigroup (i.e. binary operation * is non-associative)such that the distributive laws a * (b + c) = a * b + a * c and (a + b) * c = a * c + b * c for all a, b, c in R.

Definition 2.5. Let R be a commutative ring with one. G be any group (S any semigroup with unit) RG (RS the semigroup ring of the semigroup S over the ring R) the group ring of the group G over the ring R consists of finite formal sums of the form [n.summation over (i=1)] [[alpha].sub.i][g.sub.i], (n < [infinity]) i.e. i runs over a finite number where [[alpha].sub.i] [member of] R and [g.sub.i] [member of] G ([g.sub.i] [member of] S) satisfying the following conditions:

1. [n.summation over (i=1)] [[alpha].sub.i][m.sub.i] = [n.summation over (i=1)] [[beta].sub.i][m.sub.i] [left and right arrow] [[alpha].sub.i] = [[beta].sub.i], for i = 1, 2,..., n

2. [n.summation over (i=1)] [[alpha].sub.i][m.sub.i] + [n.summation over (i=1)] [[beta].sub.i][m.sub.i] [left and right arrow] [n.summation over (i=1)] ([[alpha].sub.i] + [[beta].sub.i])[m.sub.i]

3. ([n.summation over (i=1)] [[alpha].sub.i][m.sub.i])([n.summation over (i=1)] [[beta].sub.i][m.sub.i]) = [n.summation over (i=1)] [[gamma].sub.k][m.sub.k],[m.sub.k] = [m.sub.i][m.sub.j], where [[gamma].sub.k] = [summation][[alpha].sub.i][[beta].sub.i]

4. [r.sub.i][m.sub.i] = [m.sub.i][r.sub.i] for all [r.sub.i] [member of] R and [m.sub.i] [member of] G([m.sub.i] [membr fo] S).

5. r [n.summation over (i=1)] [r.sub.i][m.sub.i] = [n.summation over (i=1)] r[r.sub.i][m.sub.i] for all r [member of] R and [n.summation over (i=1)] [r.sub.i][m.sub.i] [member of] RG. RG is an associative ring with 0 [member of] R acts as its additive identity. Since I [member of] R we have G = IG [[subset].bar] RG and R:e = R [[subset].bar] RG where e is the identity element of G.

If we replace the group G in the above definition by a quasigroup Q we get RQ the quasigroup ring which will satisfy all the five conditions 1 to 5 given in the definition. But RQ will only be a non-associative ring without identy. As I [member of] R we have Q [[subset].bar] RQ. Thus we define quasigroup rings as follows:

Definition 2.6. For any quasigroup Q the quasigroup ring RQ is the quasigroup Q over the ring R consisting of all finite formal sums of the form [n.summation over (i=1)] [r.sub.i][q.sub.i], (n < [infinity]) i.e. i runs over a finite number where [r.sub.i] [member of] R and [q.sub.i] [member of] Q satisfying conditions 1 to 5 given in the definition of group rings above.

Note that only when Q is a quasigroup with identity (i.e. then Q is a Loop) that the quasigroup ring RQ will be a non-associative ring with unit. Here we give examples of non-associative quasigroup rings.

Example 2.2. Let Z be the ring of integers and (Q, *) be the quasigroup given by the following table:

[ILLUSTRATION OMITTED]

Clearly (Q, *) is a quasigroup and does not posses an identity element. The quasigroup ring ZQ is a non-associative ring without unit element.

Example 2.3. Let R be the ring of reals and (Q, *) be the quasigroup defined by the following table:

[ILLUSTRATION OMITTED]

(Q, *) is an idempotent quasigroup. Again RQ is a non-associative quasigroup ring without unit. Note that R<1>, R<2>, R<3>, R<4> are the subrings of RQ which are associative.

Result: All quasigroup rings RQ of a quasigroup Q over the ring R are non-associative rings without unit.

The smallest non-associative ring without unit is quasigroup ring given by the following example. This example was quoted by W.B.Kandasamy [2] as a groupoid ring.

Example 2.4. Let [Z.sub.2] = {0, 1} be the prime field of characteristic 2. (Q, *) be a quasigroup of order 3 given by the following table:

[ILLUSTRATION OMITTED]

[Z.sub.2]Q is a quasigroup ring having only eight elements given by {0, [q.sub.1], [q.sub.2], [q.sub.3], [q.sub.1] + [q.sub.2], [q.sub.2] + [q.sub.3], [q.sub.1] + [q.sub.3], [q.sub.1] + [q.sub.2] + [q.sub.3]}. Clearly, [Z.sub.2]Q is a non-associative ring without unit. This happens to be the smallest non-associative ring without unit known to us.

[section] 3. SNA-Quasigroup rings

We introduce Smarandache non-associative quasigroup rings. It is true that quasigroup rings are always non-associative. We write "Smarandache non-associative quasigroup ring" only to emphasize the fact that they are non-associative.

Definition 3.1. Let S be a quasigroup ring. S is said to be SNA-quasigroup ring (Smarandache non-associative quasigroup ring) if S contains a proper subset P such that P is an associative ring under the operations of S.

Example 3.1. Let Z be the ring of integers and Q be a quasigroup defined by the following table;

[ILLUSTRATION OMITTED]

Clearly, A = {[a.sub.0], [a.sub.1]} is group and ZQ [contains] ZA. Thus the quasigroup ring ZQ contains an associative ring properly. Hence ZQ is an SNA-quasigroup ring. Note that Q is a Smarandache quasigroup.

Example 3.2. Let R be the reals, (Q, *) be the quasigroup defined by the following table;

[ILLUSTRATION OMITTED]

Then clearly RQ is an SNA-quasigroup ring as RQ [contains] R<0, 1> and R<0, 1> is an associative ring.

Theorem 3.1. Let Q be a quasigroup and R be any ring. Then the quasigroup ring RQ is not always an SNA-quasigroup ring.

Proof. Since Q does not have an identity element, there is no guarantee that R is contained in RQ .

Example 3.3. Let R be an arbitrary ring and Q be a quasigroup defined by the table;

[ILLUSTRATION OMITTED]

Then clearly, RQ is not an SNA-quasigroup ring as the quasigroup ring RQ does not contain an associative ring.

Theorem 3.2. If Q is a quasigroup with identity, then quasigroup ring RQ is SNA-quasigroup ring.

Proof. Quasigroup with identity is a Loop. So, RI [[subset].bar] RQ and R serves as the associative ring in RQ. Thus RQ is an SNA-quasigroup ring.

Theorem 3.3. Let R be a ring. If Q is a Smarandache quasigroup, then quasigroup ring RQ is an SNA-quasigroup ring. .

Proof. Obviously RQ is a non-associative ring. As Q is a Smarandache quasigroup Q contains a group G properly. So RQ [contains] RG and RG is an associative ring contained in RQ. Therefore RQ is an SNA-quasigroup ring.

[section] 4. Substructure of SNA-quasigroup rings

Definition 4.1. Let R be a SNA-quasigroup ring. Let S be a non-empty subset of R. Then S is said to be S-quasigroup subring of R if S itself is a ring and contains a proper subset P such that P is an associative ring under the operation of R.

Example 4.1. Let Z be the ring of integers. Let Q be the quasigroup defined by the following table:

[ILLUSTRATION OMITTED]

Clearly the quasigroup ring ZQ is a non-associative ring. Consider the subset S = {1, 2, 3, 4} then S is a group and hence ZS is a group ring and hence also a quasigroup ring. Let P = {1, 2}. Note that ZS also contains ZP where P = {1, 2}. So, ZS is an S-quasigroup subring of SNA-quasigroup ring ZQ.

We have not yet been able to find a Smarandache non associative quasigroup subring for a given quasigroup ring. We think that it is not possible to obtain a subquasigroup for any quasigroup because for a quasigroup its composition table is a LATIN SQUARE.

Theorem 4.1. Let R be a quasigroup ring, if R has a SNA-quasigroup subring S, then R itself is SNA-quasigroup ring.

Proof. As S is an SNA-quasigroup surbring S contains an associative ring. As a result R contains an associtive ring. Thus R is an SNA-quasigroup ring.

References

[1] R.H.Bruck, A survey of binary systems, Springer Verlag, 1958.

[2] W.B.Kandasamy, Smarandache non-associative (SNA) rings, Smarandache Notions (book series), American Research Press, 14(2004), 281-293.

[3] D.S. Passman, The algebraic structure of group rings, Wiley- interscience, 1977.

[4] J.S.Robinson Derek, A course in the theory of Groups, Springer Verlag, 1996. Scientia Magna

Arun S. Muktibodh

Mohota Science College

Nagpur, India

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Author: | Muktibodh, Arun S. |
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Publication: | Scientia Magna |

Date: | Jun 1, 2005 |

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