Superfluous ideals of module over nearrings
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Keywords:
$N$-group, Nearring, Superfluous ideal graphAbstract
Nearrings are non-linear algebraic systems. Zero-divisor graphs based on algebraic structures like rings, module over rings are well-known. In this paper, we consider the module over a right nearring, (say, $G$). We define the superfluous ideal graph of $G$, denoted as $\mathcal{S}_{G}$. We obtain that if $G$ has DCCI, then $S_{G}$ has diameter at most 3. We characterize the set of ideals of $G$ with degree 1 in $S_{G}$ when $G$ is completely reducible. Furthermore, we prove several properties of superfluous ideal graphs which involve connectivity, completeness, etc. with explicit examples of these notions.
References
Amjadi J., The essential ideal graph of a commutative ring, Asian-European
Journal of Mathematics, 11, 4, (2018).
Anderson D.F., Asir T., Badavi A., Tamizh Chelvam T., Graphs from rings,
Springer Nature, 2021.
Archinger J., Binder F., Ecker J., Mayr P., Nobauer C., System of nearrings
and their applications, Version 2.9.1 (GAP package. 2.6, 2012).
Beck I.,Colouring of commutative rings, Journal of Algebra, 116 (1988), 208-
Bhavanari S., Kuncham S.P., Discrete mathematics and graph Theory, PHI
learning, 2009.
Bhavanari S., Kuncham S.P., Kedukodi B.S., Graph of a nearring with respect
to an ideal, Communications in Algebra, 38, 5(2010), 1957-1967.
Bhavanari S., Kuncham S.P., Nagaraju D., Prime graph of a ring, Journal of
Combinatorics, Information and System Sciences, 35 (1-2) (2010). 27-42. ISSN
-9628..
Bhavanari S., Kuncham S. P., Nearrings, fuzzy ideals, and graph theory, Chapman
and Hall, (2013), Taylor and Francis Group (London, New York), ISBN
: 9781439873106.
Bhavanari S., Goldie dimension and spanning dimension in modules and Ngroups,
In: Near-rings, Nearfields and Related topics (Review Volume), World
Scientific, Singapore, (2017), 26-41.
Cannon G.A., Neuerburg K.M., Redmond S.P., Zero-divisor graphs of nearrings
and semigroups, In: Kiechle H., Kreuzer A., Thomsen M.J. (eds) Nearrings and
Nearfields Springer, Dordrecht, 2005.
Jerzy Matczuk, Ali Majidinya, Sum-essential graphs of modules, Journal of
Algebra and Its Applications, 20, 11(2021).
Kedukodi B.S., Jagadeesha B., and Kuncham S. P., Different prime graphs of a
nearring with respect to an ideal, In: Nearrings, Nearfields and Related Topics.
World Scientific, Singapore, (2017), 185-203.
Nikmehr M.J., Nikandish R., Bakhtyiari M., On the essential graph of a commutative
ring, Journal of Algebra and Its Applications, 16, 5(2017).
Nimbhorkar S., Deshmukh V., The essential element graph of a lattice, Asian
European Journal of Mathematics, 13, 1(2020).
Pilz G., Nearrings: the theory and its applications, North Holland Publishing
Company, 23, (1983).
Rajani, S., Tapatee S.,Kedukodi B. S., Harikrishnan P. Kuncham S.P.: Superfluous
ideals of N-groups. Rend. Circ. Mat. Palermo, II. Ser (2023).
https://doi.org/10.1007/s12215-023-00888-2.
Rajani S., Kedukodi B.S., Harikrishnan P.K., Kuncham S.P.,: Essential ideal
of a matrix nearring and ideal related properties of graphs. Bol. Soc. Paran.
Mat., (accepted for publication).
Reddy Y. V. and Bhavanari S., A note on N-groups, Indian Journal Pure and
Applied Mathematics, 19, (1988), 842-845.
Reddy Y. V. and Bhavanari S., Finite spanning dimension in N-groups, The
Mathematics Student, 56, (1988). 75-80.
Sanikeh Babaei, Shiroyeh Payrovi, Esra Sengelen Sevim, On the annihilator
submodules and the annihilator essential graph, Acta Mathematica Vietnamica,
, (2019), 905-914.
Sharma P.K., Bhatwadekar S. M., A note on graphical representation of rings,
Journal of Algebra, 176, (1995),124-127.
Tapatee S., Davvaz B., Panackal H., Kedukodi B. S., Kuncham S. P., Relative
essential ideals in N-groups, Tamkang Journal of Mathematics, 54(2021).
https://doi.org/10.5556/j.tkjm.54.2023.4136.
S. Tapatee, P.K. Harikrishnan, B.S. Kedukodi, S.P. Kuncham, Graphs with
respect to superfluous elements in a lattice, Miskolc Mathematical Notes, 23,
no. 2, (2022), 929-945, doi:10.1142/S1793557120500230.
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