Certain properties of - -fuzzy subrings

Indonesian Journal of Electrical Engineering and Computer Science
Vol. 21, No. 2, February 2021, pp. 822~828
ISSN: 2502-4752, DOI: 10.11591/ijeecs.v21.i2.pp822-828  822
Journal homepage: http://ijeecs.iaescore.com
Certain properties of - -fuzzy subrings
Dilshad Alghazzawi1
, Wafaa H. Hanoon2
, Muhammad Gulzar3
, Ghazanfar Abbas4
, Nasreen Kausar5
1
Department of mathematics, King Abdulaziz University (Rabigh), Jeddah, Saudi Arabia
2
Department of Computer Science, College of Education for Girls, University of Kufa, Kufa, Iraq
3
Department of Mathematics, Government College University Faisalabad, Pakistan
4
Department of Mathematics and Statistics, Institute of Southern Punjab, Multan, Pakistan
5
Department of mathematics, University of Agriculture Faisalabad, Faisalabad, Pakistan
Article Info ABSTRACT
Article history:
Received Jun 23, 2020
Revised Aug 11, 2020
Accepted Aug 21, 2020
In this paper, we define the - -fuzzy subring and discussed various
fundamental aspects of - -fuzzy subrings. We introduce the concept of -
-level subset of this new fuzzy set and prove that - -level subset of - -
fuzzy subring form a ring. We define - -fuzzy ideal and show that set of
all - -fuzzy cosets form a ring. Moreover, we investigate the properties of
homomorphic image of - -fuzzy subring.
Keywords:
-fuzzy set
-fuzzy subring
- -fuzzy ideal
- -fuzzy set
- -fuzzy subring This is an open access article under the CC BY-SA license.
Corresponding Author:
Muhammad Gulzar
Department of Mathematics
Government College University Faisalabad, 38000, Pakistan
Email:98kohly@gmail.com
1. INTRODUCTION
In mathematics, ring theory is one of the most important part of abstract algebra. In algebra, ring
theory studies the algebraic structures of rings. Rings algebraic structure is a framework in which addition
and multiplication are well defined with some more properties.
The concept of fuzzy sets was introduced by Zadeh [1] in . Many mathematician have applied
various hybrid models of fuzzy sets and intuitionistic fuzzy sets to several algebraic structure such as group
theory [2, 3], non-associative ring [4, 5], time series [6, 7] and decision making [8]. Rosenfeld [9]
commenced the idea of fuzzy subgroups in 1971. The fuzzy subrings were initiated by Liu [10] Dixit et al.
[11] described the notion of level subgroup in . The idea of anti-fuzzy subgroups was invented by
Biswas [12]. Gupta [13] defined many classical -operators in 1991. Solairaju and Nagarajan [14] explored a
new structure and construction of -fuzzy groups in Muthuraj et al. [15] proposed the study of lower
level subsets of anti-QFS in 2010 The concept of -fuzzy normal subgroups and -fuzzy normalizer were
established by Priya et al. [16] in 2013. Sither Selvam et al. [17] used the Biswas work to modify the idea of
anti-QFNS in 2014. Alsarahead and Ahmed [18-20] commenced new concept of complex fuzzy subring,
complex fuzzy subgroup and complex fuzzy soft subgroups in . Makaba and Murali [21] discussed
fuzzy subgroups of finite groups. Rasuli [22] discussed -fuzzy subring with respect to -norm in The
-fuzzy subgroup in algebra was discussed in [23]. More development about fuzzy subgroup may be viewed

Indonesian J Elec Eng & Comp Sci ISSN: 2502-4752 
Certain properties of - -fuzzy subrings (Dilshad Alghazzawi)
823
in [24]. Shafei et al [25] studied the fuzzy logic control systems for demand management in airports and
energy efficiency by using 3D simulator.
This paper is organized as the section contains the elementary definition of -fuzzy subrings and
related results which are thoroughly crucial to understand the novelty of this article. In section , we define
the - -fuzzy subring and prove that the level subset of - -fuzzy subrings is a subring. We also define -
-fuzzy ideal and discuss its properties. In section , we use the classical ring homomorphism to investigate
the behavior of homomorphic image (inverse-image) of - -fuzzy subring.
2. PRELIMINARIES
We recall first the elementary notion of fuzzy sets which play a key role for our further analysis.
Definition 2.1. [1]: A fuzzy set of a nonempty set is a function,
, -
Definition 2.2. [10]: Let be fuzzy subset of a ring . Then is said to a fuzzy subring if
i. ( ) * ( ) ( )+
ii. ( ) * ( ) ( )+
Definition 2.3. [14]: Let and be two nonempty sets. A -fuzzy subset of set is a function
, - and
Definition 2.4. [14]: A function , - is a QFSRR of a ring if
i. ( ) * ( ) ( )+
ii. ( ) * ( ) ( )+
Definition 2.5. Let the mapping be a homomorphism. Let and be -QFSRs of and
respectively, then ( ) and ( ) are image of and the inverse image of respectively, defined as
i. ( )( ) {
* ( ) ( )+ ( )
( )
ii. ( )( ) ( ( ) )
Definition 2.6. [13]: Let , - , - , - be the algebraic product -norm on , - and is described
as * +
3. PROPERTIES OF - -FUZZY SUBRINGS
Definition 3.1. Let and be any two nonempty sets and be a -fuzzy subset of a set , any , -.
Then fuzzy set of is said to be - -fuzzy subset of (w.r.t -fuzzy set ) and defined by:
( ) * ( ) +
Remark 3.2. Clearly, ( ) ( ) and ( ) .
Remark 3.3. If and be two -fuzzy sets of . Then ( )
Definition 3.4. A -fuzzy subset of a ring is called -QFSR, and , -, if
i ( ) * ( ) ( )+
Theorem 3.5. If is a -QFSR of a ring , then
( ) ( ), for all and where is the additive identity of
Proof: Consider ( ) ( ) * ( ) ( )+
* ( ) ( )+ ( )
Hence, ( ) ( )
Theorem 3.6. If is QFSR of a ring , then is an -QFSR of
Proof: Assume that is a QFSR of a ring and and

 ISSN: 2502-4752
Indonesian J Elec Eng & Comp Sci, Vol. 21, No. 2, February 2021 : 822 - 828
824
Assume that, ( ) * ( ) + * * ( ) ( )+ +
{ * ( ) + * ( ) +} * ( ) ( )+
( ) * ( ) ( )+
Further ( ) * ( ) + * * ( ) ( )+ +
{ * ( ) + * ( ) +} * ( ) ( )+
( ) * ( ) ( )+
Consequently, is -QFSR of . In general, the converse may not be true.
Note 3.7. we take * + in all the examples
Example 3.8 Let * +, be a ring and * + Let the -fuzzy set of described by:
( ) {
Take then,
( ) * ( ) + * ( ) +
⟹ ( ) * ( ) ( )+
Further, we have ( ) * ( ) ( )+
Consequently is -QFSR of and is not QFSR of
Definition 3.9. Let be -Q-fuzzy set of universe set . For , - the level subset of - -fuzzy
set is given by:
* ( ) +
Theorem 3.10. Let is - -fuzzy subring of then is subring of for all ( )
Proof: It is quite obvious that is non-empty. Since be - -fuzzy subring of a ring which implies
that ( ) ( ) and Let then ( ) ( ) .
Now,
( ) * ( ) ( )+ * +
( ) * ( ) ( )+ * +
This implies that . Hence, is subring of
Definition 3.11. Let be a -fuzzy subset of a ring and , - Then is - -fuzzy left ideal ( -
QFLI) of if,
i ( ) * ( ) ( )+
ii ( ) ( )
Definition 3.12. Let be a -fuzzy subset of a ring and , - Then is - -fuzzy right ideal ( -
QFRI) of if,
i. ( ) * ( ) ( )+
ii. ( ) ( )
Definition 3.13. Let be a -fuzzy subset of a ring and , - Then is -QFI of if,
i. ( ) * ( ) ( )+
ii. ( ) * ( ) ( )+
Definition 3.14. Let be a -QFSR of a ring and , -. For any and , The - -fuzzy
coset of in is represented by as defined as,

825
( )( ) * ( ) + ( )
Theorem 3.15. Let be ω-QFI of ring . Then the set
* ( ) ( )+ is an ideal of ring .
Proof: Obviously because . Let be any elements.
Consider
( ) * ( ) ( )+ * ( ) ( )+
Implies that ( ) ( ). But ( ) ( )
Therefore, ( ) ( )
Implies that .
Further, let and . Consider
( ) * ( ) ( )+ * ( ) ( )+,
Implies that ( ) ( ). But ( ) ( )
Therefore, ( ) ( ).
Similarly, ( ) ( )
Implies that .
Implies that is an ideal.
Theorem 3.16. Let be an -QFI of ring , . Then,
Proof: For any , we have .
Consider,
( ) ( )( ) ( )( ) ( )
Therefore, .
Conversely, let
Implies that ( ) ( )
Consider, ( )( ) ( ) (( ) ( ) )
{ (( ) ) (( ) )}
{ (( ) ) ( )} (( ) ) ( )( )
Interchange the role of and we get ( )( ) ( )( )
Therefore, ( )( ) ( )( ), for all
Definition 3.17. Let be a -QFI of a ring The set of all - -fuzzy cosets of denoted by ⁄ form a
ring with respect to binary operation * defined by
( ) ( ) ( ) ⁄
( ) ( ) ( ) ⁄ The ring ⁄ is called
the factor ring of with respect to -QFI .
Theorem 3.18. The set ⁄ forms a ring with respect to the above stated binary operation.
Proof: Let be any
element of and
( )( ) ( ( ) )
(( ) ) (( ) ) (( ) )
(( ) ) (( ) ) (( ) )
(( ) ) ( ( ) ) ( )( )
Moreover,
( )( ) ( ( ) )

 ISSN: 2502-4752
826
* ( ) (( ) )+
But we have, (( ) ) (( ) )
(( ) ( ) ) * ( ) ) ( ( ) )+
* (( ) ) (( ) )+
* ( ) ( )+
(( ) ) ( )
( )( ) ( ) )
( )( )
Similarly, we can prove that ( )( ) ( )( )
Consequently, ( )( ) ( )( ).
Therefore is well defined. Now we prove that the following axioms of ring, for any
1) ( ) ( )
2) ( ) ,( ) ( )- , - ( ) ,
- ,( ) ( )- ( )
3) ( ) ( ) ( ) ( )
4) ( ) ( ) ( )
5) ( ) ( )
6) ( )( )
7) ( ),( )( )- , - , -
,( )( )-( )
8) ( ),( ) ( )- ( ),( ) - ( ) ( )
( ) ( ) ,( )( ) ( )( )-,
9) ,( ) ( )-( ) ,( ) -( ) ( ) ( )
( ) ( ) ,( )( ) ( )( )-
Consequently, ( ⁄ ) is a ring.
4. HOMOMORPHISM OF - -FUZZY SUBRINGS
In this section, we investigate the behavior of homomorphic image and inverse image of -QFSR.
Lemma 4.1. Let be a mapping and and be two fuzzy subsets of and respectively, then
i. ( )( ) ( ( )) ( ), for all and
ii. ( )( ) ( ( )) ( ), for all and
Proof:
(i) ( )( ) ( ( )) { ( ( )) } * ( )( ) +
( )( ) ( ( )) ( )
(ii) ( )( ) * ( ) ( ) + { * ( ) + ( ) }
{ {* ( ) ( ) + }} * ( )( ) + ( ( )) ( )
Hence, ( )( ) ( ( )) ( )
Theorem 4.2. Let be a homomorphism from a ring to a ring and be a -QFSR of ring
. Then ( ) is a -QFSR of ring
Proof: Let be a -QFSR of ring Let be any element. Then there exists unique elements
such that ( ) and ( ) and
Consider,

827
( ( )) ( ) * ( )( ) + * ( )( ( ) ( ) ) +
* ( )( ( ) ) + * ( ) + ( )
* ( ) ( )+ ( ) ( ) +
* * ( ) ( ) + * ( ) ( ) ++
* ( )( ) ( )( )+ {( ( )) ( ) ( ( )) ( )}
( ( )) ( ) {( ( )) ( ) ( ( )) ( )}
Further, ( ( )) ( ) * ( )( ) + * ( )( ( ) ( ) ) +
* ( )( ( ) ) + * ( ) + ( )
* ( ) ( )+ ( ) ( ) +
* * ( ) ( ) + * ( ) ( ) ++
* ( )( ) ( )( )+ {( ( )) ( ) ( ( )) ( )}
( ( )) ( ) {( ( )) ( ) ( ( )) ( )}
Consequently, ( ) is -QFSR of
Theorem 4.3. Let be a homomorphism from ring into a ring and be a -QFSR of ring
Then ( ) is -QFSR of ring
Proof: Let be -QFSR of ring . Let be any elements, then ( ( )) (
) ( )( ) ( ( ) )
( ( ) ( ) )
* ( ( ) ) ( ( ) )+ * ( )( ) ( )( )+
{( ( )) ( ) ( ( )) ( )}
Thus, ( ( )) ( ) {( ( )) ( ) ( ( )) ( )}
Further, ( ( )) ( ) ( )( ) ( ( ) ) ( ( ) ( ) )
* ( ( ) ) ( ( ) )+ * ( )( ) ( )( )+
{( ( )) ( ) ( ( )) ( )}
Thus, ( ( )) ( ) {( ( )) ( ) ( ( )) ( )}
Consequently ( ) is -QFSR of a ring
5. CONCLUSION
In paper, we have proved the level subset of two - -fuzzy subrings is a subring. In addition, we
have extended the study of this ideology to investigate the effect of image and inverse image of -QFSR
under ring homomorphism.
ACKNOWLEDGEMENT
This work was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University,
Jeddah, Saudi Arabia. The authors, therefore, acknowledge with thanks DSR for technical and financial
support.

 ISSN: 2502-4752
828
Conflict of interest
All authors declare no conflict of interest in this paper
REFERENCE
[1] L.A. Zadeh, “Fuzzy sets, Information and Control,” vol. no. , pp. 338-353, 1965.
[2] U. Shuaib, M. Shaheryar and W. Asghar, “On some characterization of -fuzzy subgroups,” Internationa journal of
Mathematics and Computer Science, vol. no. pp. 119-131, 2018.
[3] M. Gulzar, G. Abbas, F. Dilawar, “Algebraic properties of - -fuzzy subgroup”, International journal of
Mathematics and Computer Science, vol. 15, pp. 265-274, 2020.
[4] N. Kausar, B. U. Islam, M. Y. Javid, S. A. Ahmed, and U. Ijaz, “Characterization of non-associative rings by the
properties of their fuzzy ideals”, Journal of Taibah University for Science, vol. 13, no. 1, pp. 820-833, 2019.
[5] N. Kausar, “Direct product of finite intuitionistic fuzzy normal subrings over non-associative rings”, European
Journal of Pure and Applied Mathematics, vol. 12, no. 2, pp. 622-648, 2019.
[6] D. Shan, W. Lu and J. Yang, “The data-driven fuzzy cognitive map model and its application to prediction of time
series, “International Journal of Innovative Computing, Information and Control, vol. 14, no. 5, pp. 1583-1602,
2018.
[7] M. N. Alemu, “Fuzzy model for chaotic time series prediction”, International Journal of Innovative Computing,
Information and Control, vol. , no. , pp. 1767-1786, 2018, 2018.
[8] Y. Qin, Y. Liu, Z. Hong and H. Jia, “Hesitant fuzzy muirhead mean operations and its application to multiple
attributed decision making”, International Journal of Innovative Computing, Information and Control, vol. , no.
, pp. 1223-1238, 2018.
[9] A. Rosenfeld, “Fuzzy groups,” Journal of Mathematical Analysis and Application, vol. 35, no. 3 pp. 512-517, 1971.
[10] W. J. Liu, “Fuzzy invariant subgroups and fuzzy ideals,” Fuzzy Sets and Systems, vol. 8, no. 2, pp. 133-139, 1982.
[11] V. N. Dixit, R. Kumar, N. Ajmal, “Level subgroups and union of fuzzy subgroup,”. Fuzzy Sets and Systems, vol.
37, no. 3, pp. 359-371, 1990.
[12] R. Biswas, “Fuzzy subgroups and anti-fuzzy subgroups,” Fuzzy Sets and Systems, vol. no. , pp. 121-124, 1990.
[13] M. M. Gupta, J. Qi, “Theory of T-norms and fuzzy inference methods,” Fuzzy Sets and Systems, vol. , no. pp.
431-450, 1991.
[14] A. Solairaju, and R. Nagarajan, “A new structure and construction of - fuzzy groups” Advances in Fuzzy
Mathematics, vol. 4, no. pp. 23-29, 2009.
[15] Dr. R. Muthuraj, P. M. Sitharselvam, M. S. Muthuraman, “Anti -Fuzzy Group and Its Lower Level Subgroups,”
International Journal of Computer Application, vol. 3, no. pp. 16-20, 2010.
[16] Priya, T. Ramachandran, K. T. Nagalakshmi, “On -fuzzy Normal Subgroups” International Journal of computer
and Organization Trends, vol. no. 6, pp. 39-42, 2013.
[17] P. M. Sithar selvam, T. Priya, K. T. Nagalakshmi and T. Ramachandran, “On Some properties of anti - -fuzzy
Normal Subgroups” General Mathematics Notes, vol. 22, no. 1, pp. 1-10, 2014.
[18] M. O. Alsarahead, A. G. Ahmad, “Complex fuzzy subgroups,” Applied Mathematical Sciences, vol. no. , pp.
2011-2021, 2017.
[19] M. O. Alsarahead, A. G. Ahmad, “Complex fuzzy subrings,” International Journal of Pure and Applied
Mathematics, vol. 117, no. pp. 563-577, 2017.
[20] M. O. Alsarahead, A. G. Ahmad, “Complex fuzzy soft subgroups, Journal of Quality Measurement and Analysis,”
vol. , no. 2, pp. 17-28, 2017.
[21] B. B. Makamba, V. Murali, “A class of fuzzy subgroups of finite reflection groups,” Journal of Intelligent and
Fuzzy Systems, vol. 33, pp. 979-983, 2017.
[22] R. Rasuli, “Characterization of -fuzzy subrings (Anti -fuzzy subrings) with respect to a -norm ( -conorm),”
Journal of Information and Optimization Sciences, vol. pp. 827-837, 2018.
[23] Dr. R. Jahir Hussain, “A Review On -fuzzy subgroup in Algebra,” International Journal of Applied Engineering
Research, vol. 14, pp. 60-63, 2019.
[24] S. A. Trevijano, M. J. Chasco and J. Elorza, “The annihilator of fuzzy subgroups,” Fuzzy Sets and Systems, vol.
pp. 122-131, 2019.
[25] M. A. R. Shafei, M. A. Tawfik and D. K. Ibrahim, “Fuzzy control scheme for energy efficiency and demand
management in airports using 3D simulator,” Indonesian Journal of Electrical Engineering and Computer Science
(IJEECS), vol. no. 2, pp. 583-592, 2020.

Certain properties of - -fuzzy subrings

More Related Content

Similar to Certain properties of - -fuzzy subrings (20)

More from nooriasukmaningtyas (20)

Recently uploaded (20)

Certain properties of - -fuzzy subrings