ON NIL-SYMMETRIC RINGS AND MODULES SKEWED BY RING ENDOMORPHISM

 

Ibrahim Adnan Mustafa ^1,* , Chenar Abdulkareem Ahmed ¹

 

¹ Department of Mathematics, College of Science, University of Zakho, Zakho, Kurdistan Region, Iraq

 

*Corresponding author email: ibrahim.mustafa@uoz.edu.krd

 

ABSTRACT:

1.        INTRODUCTION

        Every ring in this study has a unique identity, and every module that is investigated is a unital module.  and  denotes the ring of integers, integers modulo  and the set of nilpotent elements in , respectively. Furthermore, denote the identity endomorphism, an endomorphism of an arbitrary ring (For short, endo) and right -module respectively.   is the left annihilator of  in . 

        A ring  is reduced (For short red-ring), if it has no nonzero nilpotent elements. However, if  implies  for , then  endo  of the  ring  is said to be  rigid (For short, -ring endo) (Krempa, 1996).  If there is a -ring endo  of ring , then  is said to be  -rigid ring (For short,  --ring) (Suarez H., et al., 2024). Note that,  --rings are red-rings by [(Hong et al., 2000), Proposition 5]. and any -ring endo of a ring is a monomorphism. Cohn introduced a ring  as reversible, if whenever , then  , for .  Lembek referred to a ring  as symmetric (For short,  -ring), if whenever , then  for According to [(Shin, 1973), Lemma 1.1], every red-ring is symmetric; however, the convers does not true in general  [(Anderson & Camillo, 1999), Example 11.5]. Although, it is clear that -rings are reversible and commutative rings are symmetric, the convers of each of them does not true in general   [(Anderson & Camillo, 1999), Example 1.5 and 11.5] and [(Marks, 2002), Example 5 and 7]. As an extension of  -rings and a specific instance of -semi-commutative rings, Chakraborty and Das presented  the idea of -symmetric rings in  (Chakraborty & Das, 2014). A ring  is right(R) (left(L)) -symmetric (For short, R(L)--ring), if for  and  then . A ring is -ring if it is both L(R) -ring.

       The concept of an -symmetric ring was first proposed by Kwak, T. K. in 2007, as an extension of -rings and a generalization of - rings. In (Kwak, 2007) an endo  of a ring  is called L(R)--symmetric ring(For short,  --ring),  if  imply for   A ring  is L(R)- --ring if there exists a L(R)- -ring endo  of  the concepts of an --ring is an extension of -rings and it is also a  generalization of - rings.

        The ring notion was recently extended to include modules. A module  is called symmetric (For short,  -module), if whenever   satisfy  then we have  ((Lambek, 1971) and (Raphael, 1975)). A module  is -semi-commutative if,  implies  for . The module  is semi-commutative if it is -semi-commutative. Buhphang and Rege in (Buhphang & Rege, 2002) examined the fundamental characteristics of semi-commutative modules. Agayev and Harmanci concentrated on semi-commutativity of subrings of matrix rings and carried out additional research on semi-commutative rings and modules in (Agayev & Harmancı, 2007).

        Motivated to the above, this article is structured to introduce and define a new kind of rings named a R--- ring as a generalization of --rings and an extension of -rings, and to explore and provide various characterizations, features and relations about this concept and to study its related properties.  Additionally, we investigate the concept of right --symmetric on some of ring extensions and localizations. This leads to a number of well-known outcomes as corollaries of our results. Then we extend the property of R-- rings to modules by introducing the notion of right --symmetric module which is a generalization of -symmetric modules and extensions of symmetric modules. We examine the characteristics of right --symmetric modules and their associated attributes, such as localizations and module extensions.

On --Symmetric Rings:

        The fundamental structure of -- rings is examined in this section, along with a number of associated ring features. We begin with the following definition.

 

Definition 2.1 An endo  of a ring  is said to be left(L)-right(R)--symmetric(For short,  L-R---ring), if whenever  for  and  A ring  is L-R- - if there exists a L-R  endo  of . Moreover,  is --ring if it is both L-R-  ---ring.

Remark 2.2:

1.     A ring  is -ring if  is --symmetric, where  is the identity endo.  

2.     Every subring  with  of an --ring is also --ring.

3.      , but the converse does not true (See (Kwak, 2007)Example 2.7(1) ).

4.     The concept of --ring is not R-L- --ring through the following example.

Example 2.3 Suppose that a ring  then

(i)     Let  be an endo defined by:

If  for  then we get  and so  since  is commutative. This yields  and hence  is R---ring. For  with  we have  and thus is not L---ring.

(ii)   Let  be an endo defined by:

By using the same technique as in (i), we may demonstrate that  is L----ring. However,  is not R----ring for  but  and thus  is not R----ring.

 

Lemma 2.4 (1) For a ring ,  is R---ring if and only if    for  and    .

       (2) Consider  be a reversible ring.  is R---ring    if and only if   is L---ring.

Proof. (1) It suffices to show that  for  and   implies , when  is right --ring.   Let , then  for   ,  and , and hence  by the condition. Thus  

(2) Let  for . If  is R---ring, then  since  is reversible, we have  and hence  is L---ring. The converse is similar.       

        The condition "is reversible" in (Proposition 2.4) is irremovable, as demonstrated by Example 2.3. While it is evident that all -symmetric objects are --ring, the following example shows that the converse is not true.

 

Example 2.5 Assume   is the ring of integer modulo  and  Using the standard addition and multiplication. Since   is --ring. Now let  be defined by . Then, for   but  and thus  is not an --ring.               

   

Consider  is a ring and ,  is called the L-annihilator of in . If , then we write  instead of  

Lemma 2.6 For a ring , then the following are equivalent for a nonzero endo :

(1)     is R---ring;

(2)     for any  and ;

(3)     if and only if   for any  and ;

(4)      for any  and

Proof. (1)  (3). Suppose that  for  and  For any  Then  and hence  Therefore  

The converse is obvious.  (1)  (2) and (3)  (4) is clear.                                   

Lemma 2.7 The class of --rings is closed under direct products.

        Proof. Note that  and  for each  Now, let  ,where Thus for  and     Since  is R---ring for each  , then  for each  So we get  Therefore, the direct product  of  is R---ring.                                       

 

Recently, it was proven that if  such that   (), then  is R(L) reversible, and the ring  is called R(L) -reversible if there exist a R(L) reversible endo  of  A ring  is -reversible (Başer et al., 2009) if it is both L(R) -reversible.

 

Theorem 2.8 Let  be a --ring. Then we have the following.

1.For , then  and Consequently,   is right -reversible ring.

2.Consider  is a monomorphism of  Then we have the following.

 i. is -symmetric ring,

ii.For  then  and   Conversely, if  or  for some  then

Proof. The proof is similar to that of [(Kwak, 2007), Theorem2.5].

EXTENSIONS OF RIGHT --SYMMETRIC RINGS :

        In this section, we investigate the properly of right --symmetric on some extensions of right --symmetric. One may ask whether the following extensions   are right --symmetric, if  is right --symmetric. According to this, many results were obtained.  Consider an  upper triangular matrix ring, matrix ring over , denoted as . Suppose that  represents the subring of  where all diagonal entries are the same.

        For any red-ring , both  and  qualify as R---rings for any given endo . However, the following counterexample demonstrates that there exists a red-ring  with an endo  such that  does not satisfy the R---rings condition.

Example 3.1 An automorphism  of  defined by:

 and

Assume  Now for  and  we have  but

Therefore,  is not --ring.

 

The trivial extension of a ring  by a -bimodule is the ring , which can be obtained by the standard addition and multiplication as follows:

This is isomorphic to the ring the usual matrix operations are used. For an endo  of a ring  and the trivial extension  of  defined by:

is an endo of . Since  is isomorphic to

The trivial extension of the red-ring is symmetric by [(Huh et al., 2005), corollary 2.4]. However, for a R---ring .  need not be a right --ring by the following example.

 

Example 3.2 Suppose the R---ring

. Assume  be an endo defined by . Take  Let

 but  Thus  is not right  --ring.

 

Proposition 3.3 Consider   is a red-ring, then  is a R---ring.

Proof. The proof is similar to that of [(Kwak, 2007), Proposition3.2].  

 

The following is an extension of the trivial extension  of the - ring to a new ring:

And,

The endo  defined by  is further extended to an endo  of a ring for any . If  is - then  is not a R---ring by [(Kwak, 2007), Example 3.4].  The following example shows that  cannot be --ring for any  even if  is an - ring.

 

Example 3.4 Consider  is an endo of an - ring . Note that  for  By [(Hong et al., 2000), Proposition 5] In particular

Let  for

But we have,

Thus  is not a R---ring.

 

Theorem 3.5 Consider  is a red-ring and . If  is a R---ring with  then  is a R---ring, where  is the ideal generated by

Proof. Suppose   If  then  If , then  is a right --ring by Proposition 3.3, Now for  the prove is similar to the proof of [(Kwak, 2007), Theorem 3.8].

 

From (Harmanci et al., 2021), Consider  is a ring and  a subring of  and  . The operations of the ring  are twice addition and multiplication. We provide sufficient and necessary criteria for  to be --ring in the following proposition.

 

Proposition 3.6 Consider  is a ring and  is a subring of  Then the following are equivalent:

(1)     is R---ring;

(2)     is R---ring.

Proof. (1)  (2) Let  with  Let ,  and  By(1),  in . Hence  and so  is R---ring,

(2)  (1) Assume that  and with  Then all components of  and  are nilpotent in  Since  is R---ring, we obtain   Hence  is R---ring.                                        

The polynomial ring over a right -symmetric is now examined to see if it is a R---ring. However, the following example shows that the answer is negative.

 

Example 3.7 Assume that  is the field of integers modulo 2, and consider  is the free algebra of polynomials with zero constant term in non-commuting intermediates    and  over  Define an automorphism  of  by :

Take an ideal  in the ring  generated by the following elements:

and  where

Now  is symmetric by [(Huh et al., 2005),Example 3.1] and so a R--ring. By [(Mohammadi et al., 2012), Example 3.6],

we have  . Now but because Hence  is not a R---ring.

 

According to Rege and Chhawchharia (Rege&Chhawchharia,1997),a ring  Armendariz exists if whenever any polynomials satisfy  then  for each  and .

Since Armendariz was the first to demonstrate that a red‑ring always satisfies this criterion, they used this  terminology ([(Armendariz, 1974), Lemma1]). Assume  is a ring with an endo  Recall that the map  by .

 

Proposition 3.8 Suppose  is an Armendariz ring then  is R---ring if and only if  is a R---ring.

Proof. It also suffices to establish necessity. Let   with  and so  for all  and .   since  is Armendariz and a R---ring. This yields  therefore,  is a R---ring.

 

Theorem 3.9 (1) For a ring   is - then  is a R---ring.

(2)                     If the skew polynomial ring  of a ring  is a -ring, then  is a  --ring.

Proof. (1) Consider  is -. Note that any - ring is reduced and  is a monomorphism by [(Marks, 2002), P.218]. We show that  is R---ring. Assume  for  Then we obtain  since  is reduced (and so symmetric). Thus,

  Since  is -,  and thus  is a R---ring.

(2) Assume  for . Let  Then  since  is -ring, we get  and so  Thus  is a R---ring.                      

        The Dorroh extension(For short DoEx) of an algebra  over a commutative ring , introduced by Dorroh in 1932(Dorroh, 1932), is a construction that enlarges  by incorporating elements of . It is defined as the Abelian group  with multiplication given by  for all  and . This operation preserves the algebraic structure while introducing a direct interaction between elements of and . Additionally, any -linear endo  of  extends naturally to an 𝑆, S-algebra homomorphism , defined by , applying  to the first component while keeping the second component fixed.

 

Theorem 3.10 Consider  is an algebra equipped with an endo  and an identity element, defined over a commutative red-ring . Then  is a R---ring if and only if the DoEx  of  by  is R---ring.

Proof. It is clear that . Since  is a commutative red-ring. Consider  and  with  Thus  Since  is --ring, we get   So Thus  is --ring.

                                                                                       

SOME LOCALIZATIONS OF RIGHT --SYMMETRIC RINGS:

        Assume that  is a monomorphism of the ring  . The construction of an over-ring of   ( A ring   is an over ring of integral domain  if  is a subring of  and  is a subring of the field of fraction  the relationship  As introduced by Jordan, is now under consideration (for more details, see (Jordan, 1982)). Define  as the subset of the skew Laurent polynomial ring  consisting of elements of the form  for  and  Notably, for  the relation  hold for any  This implies that for any  the transformation follows the pattern:

From this, it follows that   forms a subring of  equipped with the natural operation:

And,

  and

Notably,  serves as an over-ring of  and the mapping  defined by  is an automorphism of  

Jordan established that such an extension  always exists for any given pair  (Jordan, 1982).

This is achieved using left localization of the skew polynomial  with respect to the set of powers of  This extension  is commonly referred to as the Jordan extension of  by

Proposition 4.1 Consider  is a ring with a monomorphism, then  is R---ring if and only if the Jordan extension  is R---ring.

 

Proof. If  is R---ring, then so is each subring  with  Therefore, it is enough to demonstrate the necessity. Assume  is --ring and  where  for  Then  and . From  we get  and so  by assumption. Hence .

Therefore, Jordan extension  is right --ring.         

 

Recall that the map  defined by  is an endo of  and the map obviously extends  

 

Proposition 4.2 If  is an Armendariz ring, then the following claims are equivalent:

(1)   is a R---ring;

(2)   is a R---ring;

(3)   is a R---ring.

 Proof. (1)  is proven in proposition 3.8

(2)  Showing necessity is sufficient. Let  with   Then  such that and  so  Since  is R---ring, we obtain  Hence  Thus  is a R---ring.

(3)  and (3) are clear.                 

 

Proposition 4.3 Assume that  is a ring and that  is an infinite subring with all of its nonzero elements regular in . Then  is R---ring if and only if  is R---ring if and only if  is R---ring.

        Proof. It is sufficient to demonstrate that,   is --ring when so is , is obtained as the subdirect product of an infinite collection of copies of , as  comprises an infinite subring where each nonzero element is regular in  according to the hypothesis. Thus  is --ring because  is --ring by the assumption.    

ON RIGHT --SYMMETRIC MODULES:

        This section extends the idea of a R---ring to modules by introducing the notion of a right --symmetric module, which is an extension of symmetric modules and generalization of -symmetric modules. Some of the well-established results which are obtained in section 3 and section 4 are generalized to right --symmetric modules. We introduce the following definition first.

 

Definition 5.1 Assume  is a ring and  a nonzero endo of . An -module  is called a right --symmetric modules (For short R---module) if whenever  for  and  implies

Example 5.2:

1.        R---symmetric modules are exactly R---module. 

2.        For any commutative ring, any module  is an  --modules.

3.     Let  be a division ring,  and . Then  is an --module.

4.     It is clear that -symmetric modules are --module but the converse implication is not true as we see in the following example.

Example 5.3 Let  be the ring of integers. We now consider the ring  and the -module  and  an homomorphism defined on  by  where .  is R---module for  and  where  ,  we have,

Also,

But  is not -symmetric for  , , we have,

But,  

 

However, the converse is true if,  is an --module by the following Lemma.

 

Lemma 5.4 Let  be an --module, then the following are equivalent:

1.         is an  -symmetric module;

2.         is an  - symmetric module.

Proof. (1)  (2) It is clear.

(2) (1) Let  for  and . If , is trivial. Then  implies  since  is - symmetric. Hence  and since  is an --module implies that  Therefore,   is an --module and by [(Agayev et al., 2009), Theorem 2.1] is an -symmetric module.                                                   

 

Proposition 5.5 For a given endo of a ring  and an -module . The statements below are equivalent:

1.         is R---module,

2.      for any

3.      if and only if for  and

4.      for any  and

Proof. (1)  (3) Suppose that  for  and  Then  for any  and  and hence  Therefore  and . The converse is clear. (1)  (2) and (3)  (4) is obvious

 

Proposition 5.6 Suppose that  is a ring and  an endo of  and  -module. Then we have the following:

1.      implies  for each permutation  of the set  where  and .

2.      if and only if  for any

Proof. The proof is similar to the proof of [(Agayev et al., 2009), Proposition2.4].                                   

 

Proposition 5.7 Suppose  is a ring and  an endo of    and -module  Then we have the following:

1.     The class of a R---modules is closed under submodules, and direct sums.

2.     The direct product of R---modules is R---module.

3.     If  is a central idempotent of a ring  with  and , then  and  are R---module if and only if  is right --module.

Proof. (1) Depending on the definitions and algebraic structures, the proof is straightforward.

(2) Note that  and  for each . Suppose that  is --module for each  and let  where,  and  Then  for each  and  by hypothesis since  and  for each  This implies  entailing that the direct product is R---module.

(3) Establishing necessity is enough. Assume  and   are R---modules. Consider , for , and , then  And  By hypothesis, we get  and ,

 and  

 and

 is a R---module.                         

 

According to (Lee & Zhou, 2004), the module  is said to be -reduced, if for each  and each  with    

 

Lemma 5.8 ([(Raphael, 1975), Lemma 1.2]). Let  be an -module. Then the following statements are equivalent:

1.         is -reduced;

2.        The following statements are true: For each  and

a.        

b.       

c.        

If the module  is 1-red-module, it is referred to as reduced. Hence, a ring  is a red-ring if and only if   is is 1-red-module as an  -module  .

 

Proposition 5.9 Every -reduced module is a R---module.

Proof. Consider  and  with  we prove  We apply conditions of -reduced module in the process. Now  Then,

  Hence  is a R---module.

 

The following illustration shows that, in general, Proposition 5.9's converse is not true.

 

Example 5.10 Consider  denote the ring of integer modulo 4. Let the ring  and the -module  and a homomorphism  is defined by .   is R---module but not -reduced.

For, if  and  Then  but . Hence  is not -reduced.

 

Proposition 5.11 For a ring  and -module . Then the following conditions are equivalent,

i.        is R---module.

ii.            Each submodule of  is R---module.

iii.            Each finitely generated submodule of  is --module.

iv.            Each cyclic submodule of  is R---module.

Proof. It is a direct result of definitions and Proposition 3.6.

 

Theorem 5.12 Every flat module over an R---ring is an R---module.

Proof. Assume  be a flat module over the R---ring  and   a short exact sequence with  free -module. By [(Lee & Zhou, 2004), Theorem 2.3] is a R---module and we write  and any element  for  Let  where  and  Since  is flat there exists a homomorphism  such that  Now set  Then  Since  is R---module, . Then . Since  we have  Therefore  Therefore  is R---module.   

 

Proposition 5.13 Assume  are rings and  be a ring endo. If  is a right -module, then is a right -module via  for all  and   is R---module, if and only if  is R---module.

Proof. Let  be an R---module. Consider  and Such that  Then  Since  is R---module, we have,

Hence  is a --module.

Conversely.  Assume that  is onto and  is a R---module. Let  and  such that . Since  is onto, there exists  such that  and  Then  Since  is right --module, we have  Hence  Thus  is R---module.                                         

 

Now we study the -symmetric property on some module extensions and module localizations like ,,

The following concepts were introduced by Lee and Zhou. For a module , We examine  is an Abelian group under clearly addition operation. Additionally, the next, scalar product operation turns  into a right -module:

For  and

 

 becomes a right module over  as a result of these operations. In the same way, the Laurent polynomial extension  becomes a right module over  with a similar scalar product. Zhou and Lee (Lee & Zhou, 2004) also introduced notations for  module as,

. Each of the above is abelian group underneath the addition condition. Furthermore,  is a module for  under the product operation as:

In the same way, the skew Laurent polynomial module  transforms into a module on  

Again, from (Lee & Zhou, 2004), module  is known as -Armendariz  if the  below conditions holds: (i) For  and  for the case if  (ii) any  and  imply  for all  and  And then, Anderson and Camillo (Anderson & Camillo, 1999), extended the concept of Armendariz ring to Armendariz module, as follows: A -module  is Armendariz when, if  and  such that  implies  for all  and . The Armendariz property is applicable for any finite product of polynomials. Clearly,  is an Armendariz ring if and only if  is an Armendariz -module.

 

Theorem 5.14 Consider  is a -Armendariz module. Then, the statements that follow are equivalent:

1.         is R---module;

2.         is R---module;

3.         is R---module.

Proof. It suffices to demonstrate that 1  3. Let  and . Then we obtain  Let  this implies  for all . Thus, by hypothesis . Therefore  and so  is a R---module.   

Corollary 5.15 Consider  be an Armendariz module. Then the following are equivalent:

1.         is R---module;

2.         is R---module;

3.         is R---module.

Proposition 5.16 Consider  is an endo of a ring  and is -reduced module. Then is R---module over  if and only if  is R---module over  for integer

Proof.  Let  is right  --module with  where . Note that for each  and  with   Therefore, it is sufficient to display the cases  Since  The following equations are available to us:

(1)   

(2)   

(3)   

      

Since  is -reduced for any    and each -reduced module is semi-commutative. These facts are used as follows:

Eq(1) and Eq(2)  gives  and so   and  multiplying by   gives  so we have,  and  From Eq(1),(2) and (3)   we get   and,

   in a similar way. If we multiply the right side of Eq(3) by  and  respectively, then we obtain  and  in turn Inductively we assume that  where  We apply the above method to Eq. First, the induction hypotheses and Eq   give  and,

If we multiply Eq on the right side by  and  respectively, then we obtain  and so  In turn. This shows that  for all  and  with  Consequently,  for all  and  with  and thus  by [(Kwak, 2007), Theorem 2.5(1)]. This yields  and therefore  is R---module.           

 

If  implies  for , then an element  of a ring  is right regular. Regular indicates that it is both left and right regular (and so not a zero divisor), while left regular is defined similarly. Assume that  is a subset of  that is multiplicatively closed and made up of central regular elements. Let  be an automorphism of  and consider  Then  in  and the induced map  defined by   is also an automorphism.

 

Proposition 5.17 Consider a ring  and a subset  of  that is multiplicatively closed and consists of central regular elements. Then

(1)     is a R---ring if and only if is is a R---ring. 

(2)     A module is R---module if and only if  is a R---module.

Proof.(1) Assume  with  and  Since  is included in the centre of  

we have and so  for some  But  is  R---ring by the condition, so  and Hence  is R---ring.  

 (2) Since a submodule of a R---module is likewise a R---module, it is sufficient to verify the required condition. Assume that is a R---module and  for  and  where  . Since  is included in the centre of , we have  and so  By assumption . Therefore  Hence  is a R---module.     

 

Corollary 5.18 (1) For a ring  is R---ring if and only if is  a R---ring.

(2) For a -module ,  is R---module if and only if  is a R---module.

Proof (1). Consider  Then clearly  is a multiplicatively closed subset of . Since  it follows that  is right  --ring by proposition 5.17(1).

(2) It is evident from proposition 5.17(2). if . Then  is a multiplicatively closed subset of  consisting of regular central element of . Since  and                                 

 

 is a classical right quotient for  if every regular element of  is invertible in  and every element of   can be written in the form a  with  and  regular.

        A right Ore ring is a ring  where, for any  with  being regular,  with  also regular, such that  It is well known that  is a right ore ring if and only if  its classical right quotient ring  exists. Now, suppose  is a ring with the classical right quotient ring  Then any automorphism  of  extends to  by defining its action on fractions as  for all  provided that  remains regular whenever  is a regular element in

 

Theorem 5.19 Consider  is Ore ring with an endo  of  and the classical right quotient ring  ring of . Then

(1)     is a R---ring if and only if is a R---ring.

(2)       is a R---module if and only if  is a R---module.

Proof. (1) Consider  is a R---ring. Assume  and with  where  and  with  regular. Let be an  ring Then  is  and so    with  regular such that  and   Now   with  regular such that . Hence  Let  and  be the ideals in , generated by  and  within , respectively. Then each of  and  are  with    Since  is right Ore, for   with  regular such that  .Here note that  Indeed,  and so  So  

Similarly, also there exists  and  with  regular such that  Thus, we obtain that  and hence  This implies  and  So we have  It follows that  since  is a R---ring.

Similar, there exists  and  with  regular such that and, Form  We have  and hence  It follows that,

 and hence  Now we have  therefore  is a R---ring.                                                                                              

(2) Assume that  is a R---module. Let  and  with  where  and  with  regular. Let  be an  ring Then  is  and so  then   with  regular such that  and   Now   with  regular such that . Hence   Let  and  be the ideals in , generated by  and  within , respectively. Then each of  and  are  with  and   Since  is right Ore, for   with  regular such that  .Here note that  Indeed,  and so  So  

Similarly, also  and  with  regular such that  Thus, we obtain that  and hence  This implies  and  So we have  It follows that  since  is a R---module.

        Similar,  and  with  regular such that and, Form  We have  and hence  It follows that,  and hence  Now we have  therefore  is a R---module.             

CONCLUSION

        This article introduced the concept right  - symmetric rings and then extends it to right - symmetric modules, which serve as generalizations of both -symmetric rings and -symmetric modules. Several results were founded as the characterization of --symmetric rings in section 2, also for --symmetric modules in section 5. In addition to that we investigated the concept of an --symmetric rings on some of ring extensions and localizations in section 3 and 4, also for --symmetric modules in section. As a proposal for a future work, the following questions are presented;

 1. Are all right --symmetric rings and --symmetric modules necessarily non-commutative?

2. Is there a relationship between --symmetric module and -semi-commutative?

3.Are there a class of modules which are -symmetric over their endomorphism?

Acknowledgments:

        The authors would like to express their sincere gratitude to the Department of Mathematics, College of Science, University of Zakho, for providing a supportive research environment. The constructive feedback from anonymous reviewers is also gratefully acknowledged, as it significantly contributed to the improvement of this manuscript.

Author Contributions:

        Ibrahim A. Mustafa conceptualized the study and drafted the initial version of the manuscript. Chenar A. Ahmed contributed to the development of the theoretical framework, reviewed the manuscript critically, and assisted in its final revision. Both authors read and approved the final version of the manuscript.

Declaration:

        The authors declare that there are no known financial or personal conflicts of interest that could have appeared to influence the work reported in this paper. The manuscript has not been published elsewhere and is not under consideration by any other journal.

Funding:

        This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

REFERENCES

Agayev, N., Halıcıoğlu, S., & Harmancı, A. (2009). On symmetric modules. Riv. Mat. Univ. Parma, (8) 2,91-99.

Agayev, N., & Harmancı, A. (2007). On semicommutative modules and rings. Kyun. Math. Jour. 47, 21-30.

Anderson, D. D., & Camillo, V. (1999). Semigroups and rings whose zero products commute. Communications in Algebra, 27(6), 2847–2852. https://doi.org/10.1080/00927879908826596.

Armendariz, E. P. (1974). A note on extensions of Baer and PP-rings. Journal of the Australian Mathematical Society, 18(4), 470–473. https://doi.org/10.1017/S1446788700029190

Başer, M., Hong, C. Y., & Kwak, T. K. (2009). On extended reversible rings. Algebra Colloquium,  16(01), 37–48. https://doi.org/10.1142/S1005386709000054

Buhphang, A. M., & Rege, M. B. (2002). Semi-commutative modules and Armendariz modules. Arab. J. Math. Sci, 8, 53–65.

Chakraborty, U. S., & Das, K. (2014). On Nil‐Symmetric Rings. Journal of Mathematics, 2O14(1),483784.  https://doi.org/10.1155/2014/483784.

Cohn, P. M. (1999). Reversible rings. Bulletin of the London Mathematical Society, 31(6), 641–648. https://doi.org/10.1112/S0024609399006116.

Dorroh, J. L. (1932). Concerning adjunctions to algebras.

Harmanci, A., Kose, H., & Ungor, B. (2021). Symmetricity and reversibility from the perspective of nilpotents. Communications of the Korean Mathematical Society, 36(2), 209–227. https://doi.org/10.4134/CKMS.c200209

Hong, C. Y., Kim, N. K., & Kwak, T. K. (2000). Ore extensions of Baer and pp-rings. Journal of Pure and Applied Algebra, 151(3), 215–226. https://doi.org/10.1016/S0022-4049(99)00020-1

Huh, C., Kim, H. K., Kim, N. K., & Lee, Y. (2005). Basic examples and extensions of symmetric rings. Journal of Pure and Applied Algebra, 202(1–3), 154–167. https://doi.org/10.1016/j.jpaa.2005.01.009

Jordan, D. A. (1982). Bijective extensions of injective ring endomorphisms. Journal of the London Mathematical Society, 2(3), 435–448.   https://doi.org/10.1112/jlms/s2-25.3.435

Krempa, J. (1996). Some examples of reduced-rings. Algebra Colloq, 3(4), 289–300.

Kwak, T.-K. (2007). Extensions of extended symmetric rings. Bulletin of The Korean Mathematical Society, 44(4), 777–788. https://doi.org/10.4134/BKMS.2007.44.4.777

Lambek, J. (1971). On the representation of modules by sheaves of factor modules. Canadian Mathematical Bulletin, 14(3), 359–368. https://doi.org/10.4153/CMB-1971-065-1

Lee, T.-K., & Zhou, Y. (2004). Reduced modules. Rings, Modules, Algebras and Abelian Groups, 236, 365–377.

Marks, G. (2002). Reversible and symmetric rings. Journal of Pure and Applied Algebra, 174(3), 311–318. https://doi.org/10.1016/S0022-4049(02)00070-1

Mohammadi, R., Moussavi, A., & Zahiri, M. (2012). On nil-semicommutative rings. International Electronic Journal of Algebra, 11(11), 20–37.

Raphael, R. (1975). Some remarks on regular and strongly regular rings. Canadian Mathematical Bulletin, 17(5), 709–712. https://doi.org/10.4153/CMB-1974-128-x

Rege, M. B., & Chhawchharia, S. (1997). Armendariz rings. Proc. Japan Acad. Ser. A Math. Sci. Proc. Vol. 73(A).

Shin, G. (1973). Prime ideals and sheaf representation of a pseudo symmetric ring. Transactions of the American Mathematical Society, 184, 43–60. https://doi.org/10.1090/S0002-9947-1973-0338058-9

Agayev N., Halicioğlu S. & Harmanci A. (2009). On Reduced Modules. Commun. Fac. Sci. Univ. Ank. Series, 1 , 9-16.

Suarez H., Higuera S. & Reyes A. (2024). On -skew Reflexive-Nilpotents-Property for Rings. Algeb. Disc. Math. v. 37, N 1, pp. 134-159. https://doi.org/10.48550/arXiv.2110.14061

Hassan, R. M., & Awla, S. H. (2025). SMARANDACHE ANTI ZERO DIVISORS. Science Journal of University of. Zakho. 13(1). 52-57. https://doi.org/10.25271/sjuoz.2025.13.1.1342

Haso, K. S., & Khalaf, A. B. (2022). On Cubic Fuzzy Groups and Cubic Fuzzy Normal Subgroups. Science Journal of University of Zakho, 10(3), 105–111. https://doi.org/10.25271/sjuoz.2022.10.3.907