^{1}

^{*}

^{2}

^{*}

Let us call a ring
R (without identity) to be right symmetric if for any triple
*a*,
*b*,
*c*,∈
*R*
abc = 0 then acb = 0. Such rings are neither symmetric nor reversible (in general) but are semicommutative. With an idempotent they take care of the sheaf representation as obtained by Lambek. Klein 4-rings and their several generalizations and extensions are proved to be members of such class of rings. An extension obtained is a McCoy ring and its power series ring is also proved to be a McCoy ring.

A ring

A weaker notion of symmetric is reversible which P.M. Cohn defined in [

with 1) satisfies

For a ring

On the other hand one sees that if

In Section 2, after definition, we gave some examples of right symmetric rings which are not symmetric, and developed some interactions with other classes of rings such as von Neuman regular, semicommutative, and Armendariz. In Section 3 we did some extensions of Klein 4-rings and a McCoy ring is constructed in the last section.

One notices that, in a ring without 1, and with

Definitions 2.1. A ring

Examples 2.2. (1) Klein 4-rings. ([

Let us consider all possible products of the three non-zero elements of V. There are total

Similarly, the opposite ring,

(2) For any ring

The converse is obvious. Analogously,

we let

but

Similarly,

(3) Let

(4) In [

There is a symmetry between right and left symmetric rings, because a ring

A ring R is said to be semicommutative as defined by Bell in [

A right symmetric ring in general is non abelian, non duo, non reflexive, and not a von Neumann regular ring. We pose quick counter examples for these claims. The Klein 4-ring

It is defined in [

There are several right symmetric rings without one which are symmetric. For instance, the ring of strictly upper triangular matrices over any ring is without one and is symmetric. Few more cases are given in the following:

Proposition 2.3. (1) Every symmetric ring is right symmetric and every right symmetric ring with one is sym- metric.

(2) Every reduced ring is right symmetric ( [

(3) Every right symmetric ring is semicommutative.

(4) Every von Neumann regular ring which is right symmetric is symmetric.

(5) Every reversible ring which is right symmetric is symmetric.

(6) Every ring with involution which is right symmetric is symmetric.

(7) Every ring with a reversible involution is right symmetric and hence symmetric.

(8) (1) - (7) all hold if we replace right by left.

Proof: (1) and (5) are obvious.

(2) Let a ring

Conversely, let

(3) Let

(4) Assume that

(6) Let

(7) Let R be a ring with an involution

(8) holds by left and right symmetry. □

A quick consequence of Proposition 2.3 (6) is the following.

Corollary 2.4. Every right symmetric ring which is not symmetric cannot adhere to an involution.

Examples 2.5. Hence,

2.6. Some minimalities: (1)

(2) Next higher order noncommutative rings are of order eight. So two minimal noncommutative symmetric rings are strictly upper and lower triangular matrix rings

(3) ( [

(see details in [

Reappearance of the Lambek Criterion: Lambek proved in [

Proposition 2.7. A ring

Proof: “Only if”, is obvious, because a symmetric ring with 1 is a right symmetric ring with an idempotent. For “if”, consider that

A ring R is called Armendariz as introduced by Rege, S. Chhawchharia in [

The first part of the following lemma is proved by Nielsen in ( [

Lemma 2.8. Let R be a right symmetric ring. Let

Theorem 2.9. Let R be a right symmetric ring. Let

in

Proof: Assume that the coefficients of

Now the coefficients in

multiply the remaining sum by

Corollary 2.10. Let

(1)

(2) For every pair of polynomials

(3) For every pair of polynomials

Proof: (1) and (2) are followed from Theorem 2.9.

(3) Let

Now we pose few more examples of one sided symmetric rings. These rings are extensions of

Theorem 3.1. For any prime

Proof: It is known that up to isomorphism there are eleven rings of order

and its opposite ring

Both rings are of characteristic

Hence

Clearly

Let

These rules clearly imply that

Theorem 3.2. The ring

Proof: Assume that

Assume that

Note that for any

Same will be the consequences if we replace

and

So

On the other hand, assume that

The second part can be obtained by symmetry. □

Trivial extension of a ring: Let

Theorem 3.3. The trivial extension ring

Proof: In short we write

(a)

We want to prove that

(c)

As in Example 2.2, (a) holds if either

If

Remarks 3.4. (1)

but

(2) It is known that if

(3) The ring of

and

Thus, in general,

Theorem 3.5. For a commutative indeterminate

Proof: Because _{i}, b_{j},

where

Also assume that

where

We want to prove that if

where

(I) We have five options for

(II) Let

(III) Let us have

(IV) Now assume that

The rest are similar.

Definitely, we need to watch the situation for non-zero values, for instance, if we let

In [

In ( [

In next result we prove that the right symmetric ring

Theorem 4.1. The ring

Proof: Again, let

by these symbols and define the multiplication on

Then

the form_{k} be distinct so that

Assume that

On the other hand, note that

of the form

is left McCoy. Hence

Remarks 4.2. It follows from above that

(i)

(ii)

Example 4.3. In Section 3 of [

McCoy, is also McCoy. A typical element of

where

the other hand, let

the set

We end up at a general statement. The following corollary can be proved by the methods used in Theorem 4.1.

Corollary 4.4. The power series ring

This project was funded by the ISR, Umm Al-Qura University, under Grant No: 43305007. The authors, therefore, acknowledge with thanks the financial and technical support provided by ISR.