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The Scientific World Journal
Volume 2013, Article ID 650702, 8 pages
http://dx.doi.org/10.1155/2013/650702
Research Article

Reversible Rings with Involutions and Some Minimalities

Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 12 September 2013; Accepted 24 November 2013

Academic Editors: A. Badawi and Y. Lee

Copyright © 2013 W. M. Fakieh and S. K. Nauman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In continuation of the recent developments on extended reversibilities on rings, we initiate here a study on reversible rings with involutions, or, in short, *-reversible rings. These rings are symmetric, reversible, reflexive, and semicommutative. In this note we will study some properties and examples of *-reversible rings. It is proved here that the polynomial rings of *-reversible rings may not be *-reversible. A criterion for rings which cannot adhere to any involution is developed and it is observed that a minimal noninvolutary ring is of order 4 and that a minimal noncommutative *-reversible ring is of order 16.

1. Introduction

Throughout this note we assume that rings are associative may be without identity. We will specifically mention if a ring is with the identity. In most of the cases the rings are equipped with an involution that we refer to by .

A ring is termed as a reversible ring by Cohn in [1] if for any pair of elements , then . Anderson and Camillo used the notation for the same type of rings in [2]. Recently, the notion of reversibility is extended to -reversibility in [3] and strong -reversibility in [4], where is an endomorphism. Thus, if , such that , then is termed as right reversible in [3] and if the converse holds in the sense that , then is termed as right strong -reversible in [4]. The ring is called right -reversible and right strong -reversible, respectively. Analogously, the terms -reversible and strong -reversible are defined.

In this note we replace the endomorphism by an involution which is an anti-automorphism on of order two. Thus, the -reversibility is replaced by -reversibility that will be defined in Section 2. Note that neither -reversibility is -reversibility nor -reversibility is -reversibility, because clearly, in general, an anti-automorphism cannot be an endomorphism, and, conversely, an endomorphism cannot be an anti-automorphism. Though some results of these notions may go parallel, we will work on -reversibility from scratch.

A ring is zero ring if and a domain if is a no-zero-divisors ring (see [5]), that is, without non-zero zero divisors. is called reduced if has no non-zero nilpotent elements and symmetric if for any triple , then , where is a permutation. Symmetric rings were introduced by Lambek in [6]. In [2], the notation is used for a symmetric ring. It is known that every reduced ring is symmetric [2, 6] and neither a symmetric ring is reversible nor a reversible ring is symmetric (for this debate and examples and counterexamples, see [2, 79]). For a ring with identity, it is clear that every symmetric ring is reversible. All these types of rings are semicommutative, where a ring is semicommutative, in the sense of Bell [10], if for any pair of elements , for all . Semicommutative rings have many names in the literature. For details and examples and counter examples, see [11].A ring is reflexive if for any pair of elements then Symmetric and reversible rings are reflexive.

In Section 2, we have given some properties and several examples and counter examples related to -reversible rings. In Section 3, we have modified an example [12, Example 2] to verify that the polynomial rings of -reversible rings may not be -reversible. Section 4 deals with identifying minimal left or right symmetric, symmetric, reflexive, reversible, and -reversible rings. In particular, a criterion is developed for rings which are noninvolutary; that is, a ring which cannot adhere to involutions.

For elementary notions about rings we refer to [13, 14] and for rings with involution to [1518].

2. -Reversible Rings

Definitions 1. Let be a ring with the involution . We say that an element is right -reversible if there is a non-zero element, such that , then . Similarly, let us call an element right -inverse-reversible if there is a non-zero element , such that , then . Analogously, we define left -reversible, -reversible, left -inverse-reversible, and -inverse-reversible elements.
If all elements of the ring are right (left, two-sided) -reversible or -inverse-reversible, then we use the same term for the ring .

Proposition 2. For any ring with the involution , the following are equivalent:
(1) is right -reversible,
(2) is left -reversible,
(3) is right -inverse-reversible,
(4) is left -inverse-reversible.

Proof. Let for any pair of non-zero elements and and annihilate each other in the direction , which implies by definition that and then again . Hence, finally, . The rest can be proved analogously.

In the light of the above proposition if all elements of a ring which annihilate each other are left (or right) -reversible (or left or right -inverse-reversible), then they are -reversible. Hence, we have the following.

Definition 3. If satisfies any one of the conditions of Proposition 2, then we say that is a reversible ring with the involution or that is a -reversible ring.

Example 4. All domains with some involution are -reversible. For instance, the commutative ring is reversible with the involution defined by . The ring of real quaternions is reversible with the natural involution defined on its elements by .
Among the nondomains, if, for some involution , a domain is -reversible, then the cartesian product under the induced involuton is -reversible, where the induced involution on is defined by for all .
Consider the product which is a commutative ring under usual multiplication. Let be an involution on defined by , for all . Now, if we let ; then we see that while . Hence, is not -reversible. Clearly, is reduced. Hence, one concludes that a ring with some involution may be commutative and reduced but not -reversible.
Note that is not reduced but is reversible with the trivial involution.
For any ring the ring of strictly upper triangular matrices (or strictly lower triangular matrices ) is not -reversible for any involution on (see details in Example 23).
An involution on a ring is an anisotropic involution if there exists no , such that ; otherwise it is called isotropic [17, 19]. Let us say that a ring is anisotropic (isotropic) if it adheres to an anisotropic (isotropic) involution.
For example, the rings and in Example 4(1) are anisotropic.
On the other hand, the noncommutative quaternion algebra (see [17, page 25]) over any field with and with a basis , and with a natural involution defined by may not be -reversible. For instance, if , and then but .
The group ring (where is the group of real quaternions), as discussed in [8, Example 7], is reversible and is not symmetric. Let us define an involution on its elements by Then if and only if . This holds even though . For instance, if , then one calculates that . Hence, is not -reversible and it is isotropic under .
See [19, Example 2]. Consider the group ring , where is the symmetric group on three letters. adheres to an involution defined by . Assume that , and . Then and are anisotropic and -reversible while is isotropic and not -reversible, so the ring is isotropic and not -reversible.
Examples of left or right -reversible elements of a ring. In Proposition 2, though it is determined that a one-sided -reversible or -inverse-reversible ring is just -reversible, this criterion does not hold for individual elements. For example, consider the ring of matrices over any ring , with or without . Then is an involution ring with the involution , where, if , then is the transpose of . If , then is a symmetric matrix.
Now, let such that is symmetric and that . Then . Hence, is right -reversible.
Note that, a right (or left) -reversible element may not be a left (or right) -reversible. For instance, in case of elementary matrices in , , but . So is not -reversible, but it is anisotropic, because, for all , .

A criterion for the equality of different -reversible elements is the following.

Proposition 5. For any reversible ring with the involution and for any element , the following are equivalent:
(1) is right -reversible,
(2) is left -reversible,
(3) is right -inverse-reversible,
(4) is left -inverse-reversible.

Proof. Proofs can be obtained directly by Definitions 1.

By the above proposition we conclude that if is reversible, then any left or right -reversible or -inverse-reversible element is simply termed as a -reversible element.

If a ring is -reversible, then we have the following.

Proposition 6. Every -reversible ring is (1) symmetric, (2) reversible, (3) reflexive, and (4) semicommutative.

Proof. Let be a -reversible ring. Assume that for any , then or or that . Using the double involutions we get .
Again, means that or that implies .
Using , by the similar arguments as above, . Finally, by symmetry we get and . Hence, is symmetric.
It is proved in the first line of the proof of Proposition 2.
and These are obvious.

For any subset of a ring , the right and left annihilators of in are denoted by and , respectively. In particular, if , then we use the terminologies and .

Let be an involution ring with the involution . Because a -reversible ring is semicommutative, so every left or right annihilator of is an ideal (see [20, Lemma 1.1]). We restate here Proposition 2.3 of [4] in terms of -reversible rings which provides some additional information. The proof of each equivality is elementary.

Proposition 7. Let be an involution ring with an involution . Then the following are equivalent:
is a -reversible ring;
for each element ;
for each element ;
for each element ;
for each element ;
replace and by subsets and of , respectively, in ;
for any two non-empty subsets and of if and only if if and only if .

An element is called symmetric with respect to if .

Proposition 8. Let be a ring with 1 and with an involution .
(1) If is -reversible, then every idempotent in is symmetric with respect to . In particular, .
(2) Let be a central idempotent. Then and are -reversible if and only if is -reversible.
(3) is -reversible if and only if, for every central idempotent is -reversible.
(4) Let be abelian (i.e., every idempotent of is central). Then is -reversible if and only if, for every idempotent is -reversible.

Proof. Indeed, if , then implies that , or. Hence, in particular, .
Let be -reversible. Let be an idempotent. Then by , and so and are -subrings of . Hence, these are -reversible.
Conversely, let and be -reversible. Let for some . Then . So, by hypothesis, .
Again by hypothesis, which implies that Hence, we conclude that .
and follow from .

Proposition 9. Let and be rings and let be an isomorphism. Then we have the following.
(1) - is an involution on if and only if is an involution on .
(2) An element is right (left, two-sided) -reversible if and only if its image is right (left, two-sided) -reversible.
(3) An element is right (left, two-sided) -inverse-reversible if and only if its image is right (left, two-sided) -inverse-reversible.
(4) is -reversible if and only if is -reversible.
(5) is -inverse-reversible if and only if is -inverse-reversible.

Proof. Clearly, is an anti-automorphism on of order two and conversely is an anti-automorphism on of order two. It is a routine work to check that if is an involution on then is an involution on and conversely if is an involution on then is an involution on .
Let be right -reversible and for some . Then . The image of in is , and, naturally, the image of in is . Thus, in which both products are non-zero and this implies that Hence, is right -reversible.
Conversely, if is -reversible, then there is , such that which gives But there exist and such that and . So and This means that which implies that Hence, is right -reversible.
The proofs of the remaining parts of and that of ,, and are analogous.

Central idempotents play important role in a direct sum decomposition of a ring. If is a ring with involution and is a direct summand of , then can be given a structure of an involution ring [15, Lemma 2.3] and the converse also holds naturally. Then it follows from Propositions 8 and 9 and from Section 7 of [13] the following.

Theorem 10. All direct sums and direct summands of -reversible rings are -reversible.

Examples 11 (trivial extensions). Let be any ring; a trivial extension of is a subring of the upper triangular matrix ring over and is defined as
Define an involution on the ring where is a prime, by Clearly, is -reversible.
Note that for any prime is reduced. If we replace by a composite number such that is not reduced, then may not be -reversible. For instance, one can check that is not -reversible; however, is -reversible. We further have the following.

Theorem 12. If has an involution , then the involution on defined by is an involution. If is reduced and -reversible, then the trivial extension is -reversible.

Proof. It is a routine work to check that is an involution on
Assume that is reduced and -reversible. Let Then By the -reversible property, . Moreover, because every reduced ring is symmetric and semicommutative, so by (15) which implies that . Then which gives or that . Again by (15) and , we get ; hence, . Combining these we conclude that

The Dorroh Extension. Let be an algebra over a commutative ring . The Dorroh extension of by is a ring in which sum of elements is defined componentwise and the product is defined by the rule

If the algebra adheres to an involution , then an induced involution on is for every . We prove the following.

Theorem 13. Let , be an integral domain and an algebra with over . Then with an involution is -reversible if and only if its Dorroh extension is -reversible.

Proof. Let and be as given in the hypothesis with as -reversible. Consider two non-zero elements of such that . Then we prove that . Clearly, implies that either or . Assume first that . Then and so where (see Proposition 8) and (by the definition of an involution on an algebra over a ring). Thus, we get which means that same conclusion can be obtained if we take . The converse is obvious.

3. The Polynomial Rings of -Reversible Rings

Note that if a ring is commutative, reduced, or Armendariz, then so is But if is semicommutative, reversible, or symmetric, then may not be either semicommutative, reversible, or symmetric. These were established in [12, Example 2], [20, Example 2.1], and [21, Example 3.1], respectively, by providing the same counterexample. We continue this example to prove that if is -reversible with some involution , then may not be -reversible. To get the goal we will make some modification in the example.

Example 14. A noncommutative ring with (or without) identity is symmetric and reversible but not -reversible with some involution .

Let , and be some mutually noncommutative indeterminates and let be a free polynomial algebra. Consider the ring , and define an ideal generated by the expressions where . The factor ring is reversible [20, Example  2.1] and symmetric [21, Example  3.1].

Define an involution on by , and ; for all and . This is a routine work to check that this is an involution on and that and obviously Define the induced involution on by setting , for all . Now , but . Hence, is not -reversible.

For the without identity part, one may notice that, in seems to be a superfluous part just to bring the identity in the system. is still an ideal of and the ring satisfies all claims as stated in [20, Example 2.1] and [12, Example 3.1]. The involution can be defined by setting and , and the induced involution on can be obtained as previously done. With this involution is not -reversible.

Note that or just are domains, so these are symmetric, reversible, and -reversible, but the factor rings and are not.

Example 15. A noncommutative ring is symmetric, reversible, and -reversible for some involution .

We make some modification in Example 14. Let the ring be as in Example 14, and let the ideal be generated by the set where .

Theorem 16. (i) The factor ring or is (15) reversible, (2) symmetric, (3) semicommutative, and (4) -reversible for some involution .
(ii)or does not satisfy any of , or .

Proof. If (4) holds, then , (2) and , by default, followed from Proposition 6. We only prove (4) for ; such a proof for follows automatically.
First define the involution on and the induced involution on as in Example 14. Clearly, and
So, for (4), assume that is an involution on which is induced from . As previously stated, in [12, 20, 21], let us call each product of the indeterminates a monomial and say that is a monomial of degree if it is a product of exactly number of these indeterminates. Let be the set of all linear combinations of such monomials of degree is finite and the ideal is homogeneous in the sense that if , with , then every .
We will prove first thatif with , then . The different cases for this situation are as under
First four are identical; so, if then we see that . The same holds for the remaining.
Second four are also identical: so if then we see that Same holds for the remaining.
For the second last we have: then, and the same holds for the last one.
Now let , such that . Then we will prove that Assume that and . Then , where all monomial in are of degree , so By the above , so Then Again, from the above and as , we only need to prove that With the option , let us pick and where is some monomial. Then
For the option , let us pick and . Then Similarly, the remaining options and the possible combinations such that can be simplified to prove that . Hence, we conclude that is -reversible.
(ii) The common argument which everyone poses is the following. Let . Then but . So, is not semicommutative. Hence, it is neither reversible nor symmetric. There is no question of -reversibility as well. The same holds for .

A ring is Armendariz, as introduced in [22], if, for any commuting indeterminate , the polynomials are such that , and if and , then For instance, if is reduced, then is Armendariz. A favourable conditional case is the following.

Theorem 17. If is an Armendariz ring, then is -reversible if and only if is -reversible under the induced involution defined by , for every polynomial .

Proof. If part is trivial.
For only if, let be Armendariz and is -reversible, and let for some . Then for any pair of coefficient and , we have , which implies that with and . Hence, we plainly get .

4. The Story of Some Minimalities

In [9] a ring is defined to be right (respectively left) symmetric if for any triple , then (respectively ). If , then every right (or left) symmetric ring becomes symmetric, which returns the original definition of Lambek [6] of a symmetric ring.

Clearly, all commutative rings are left or right symmetric, symmetric, reversible, reflexive, duo (every right or left ideal is an ideal), semicommutative, and at least have a trivial involution. In this section, first we have obtained a criterion for rings to be noninvolutary and then we will find minimal (cardinality-wise) right and left symmetric, symmetric, reversible, reflexive, noninvolutary, and -reversible noncommutative rings.

The following theorem is a criterion for rings to be noninvolutary.

Theorem 18. A right (or left) symmetric ring which is not symmetric cannot adhere to an involution.

Proof. Let be a right symmetric ring which is not symmetric. Assume on contradiction that adheres to an involution . If for some , then, because is right symmetric, . Then or . Doubling the involution gives which means that . Again, gives and by the doubling of involution one gets and so the right symmetry gives . Hence, we conclude that is symmetric, which is a clear contradiction. Similarly, one can prove that if is left symmetric and it is not symmetric, then it cannot have an involution.

One can deduce from above the following:

Corollary 19. A right (or left) symmetric ring with an involution is symmetric.

Example 20. Consider the, so called, Klein-4 ring which is a Klein 4-group with respect to addition. The characteristic of this ring is and the relations among its elements are (see [8, Example 1]. Erroneously it is considered symmetric there. This ring is not symmetric, simply because but . Similarly, , but Hence, this ring is right symmetric only. By Theorem 18, it is clear that is not agreed to adhere to any involution.
The same is the case for which is left symmetric but not right symmetric and so is not symmetric. Hence, is also free from any involution.
Under these situations there is no question of -reversibility on and .
Note that, up to isomorphism, the only noncommutative rings of order four are and . Hence, and are the smallest (up to isomorphism) noncommutative right and left symmetric rings (as in Example 20), respectively. These rings are the smallest nonreduced (because is nilpotent), nonsymmetric, nonreversible, nonreflexive (, but ), nonabelian (), nonduo ( is a right ideal of which is not an ideal) and noninvolutary, so they are not -reversible as well.

Example 21. The ring of strictly upper triangular matrices over , namely, has only eight elements. It is noncommutative and is clearly symmetric, and, for the same reason, it is reflexive. But it is not reversible, because but . This ring is minimal with such properties. It adheres to an involution defined on its elements by but the ring is not -reversible. In fact it is not -reversible for any involution on it, because it is not reversible (see Proposition 5).
The claim that it is minimal noncommutative symmetric and reflexive is clear, because all rings of order less than eight are commutative other than the two rings of order four, namely and , which we already have proved that are neither symmetric nor reversible. Hence, we conclude the following.

Theorem 22. A minimal noncommutative reflexive and symmetric ring is of order eight and is isomorphic to the ring of strictly upper triangular matrices over , namely, . This ring is neither -reversible nor reversible.

The same holds for the ring of strictly lower triangular matrices over , namely, .

The above rings are without one; for a ring with one we have a different minimal situation.

Example 23. [21, Example 2.5 and Theorem 2.6] states that if a ring with identity is a minimal noncommutative symmetric ring, then is of order 16 and is isomorphic to the ring ; especially, is a duo ring, where Moreover, in [23, Theorem 5] it states that if (with identity) is a minimal noncommutative reflexive ring, then is a ring of order 16 such that is isomorphic to when is abelian and to when is nonabelian.
We will prove that is a minimal -reversible ring. For this we only prove that is -reversible under some involution . The rest follows from [21, Example 2.5 and Theorem 2.6], [23, Theorem 5] and Proposition 6.
Let us define an involution on the elements of by Note that because , for all .
This is an involution on . Indeed,
We claim that this involution is -reversible.
Note that is local and its only nontrivial ideal is its Jacobson radical so all other elements outside this maximal ideal are units. Thus, for any pair of non-zero elements if and only if and both belong to ; otherwise . Hence, is a zero ring and so implies that Hence, the claim is confirmed. This ring is obviously reversible as well. Hence, the following is proved.

Theorem 24. A minimal noncommutative -reversible ring is of order sixteen and is isomorphic to the ring .

Example 25. A ring is reversible but neither symmetric nor -reversible.

Consider the group ring where is the group of quaternions. It is discussed in detail in [8, Example 7] that this group ring is reversible but not symmetric. A natural involution induced on is the involution on defined by , for all . In Example 4 it is verified that is not -reversible. In fact, is not symmetric, so it cannot be -reversible for any involution (by Proposition 6). The order of the ring is 256.

A Comment on an Open Problem by Marks. Marks in [8] posed a problem that whether there is any ring with the identity which has smaller size than and is reversible but not symmetric. We add here our comment that such a ring is not reversible with any involution as well (by Proposition 6).

Acknowledgment

This work was funded by the Deanship of Scientific Research (DSR) of King Abdulaziz University, Jeddah, under Grant no. 130-067-D1433. The authors, therefore, acknowledge with thanks the technical and financial support provided by DSR.

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