Geodesic
On the finite Goldie dimension of sum of two ideals of an R-group
Discussiones Mathematicae. General Algebra and Applications, Tome 43 (2023) no. 2, pp. 177-187.

Voir la notice de l'article provenant de la source Library of Science

We consider an R-group G, where R is a zero symmetric right nearring. We obtain the Ω-dimension of sum of two ideals of G, as a natural generalization of sum of two subspaces of a finite dimensional vector space; indeed, difficulty due to non-linearity in G. However, in this paper we overcome the situation under a suitable assumption. More precisely, we prove that for a proper ideal Ω of G with Ω-finite Goldie dimension (Ω-FGD), if K_1, K_2 are ideals of G wherein K_1∩ K_2 is an Ω-complement, then dim_Ω(K_1+K_2)=dim_Ω(K_1)+ dim_Ω(K_2)-dim_Ω(K_1∩ K_2). In the sequel, we prove several properties.
Keywords: nearring, essential ideal, uniform ideal, finite dimension 
@article{DMGAA_2023_43_2_a0,
     author = {Sahoo, Tapatee and Kedukodi, Babushri Srinivas and Harikrishnan, Panackal and Kuncham, Syam Prasad},
     title = {On the finite {Goldie} dimension of sum of two ideals of an {R-group}},
     journal = {Discussiones Mathematicae. General Algebra and Applications},
     pages = {177--187},
     publisher = {mathdoc},
     volume = {43},
     number = {2},
     year = {2023},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGAA_2023_43_2_a0/}
}
TY  - JOUR
AU  - Sahoo, Tapatee
AU  - Kedukodi, Babushri Srinivas
AU  - Harikrishnan, Panackal
AU  - Kuncham, Syam Prasad
TI  - On the finite Goldie dimension of sum of two ideals of an R-group
JO  - Discussiones Mathematicae. General Algebra and Applications
PY  - 2023
SP  - 177
EP  - 187
VL  - 43
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMGAA_2023_43_2_a0/
LA  - en
ID  - DMGAA_2023_43_2_a0
ER  - 
%0 Journal Article
%A Sahoo, Tapatee
%A Kedukodi, Babushri Srinivas
%A Harikrishnan, Panackal
%A Kuncham, Syam Prasad
%T On the finite Goldie dimension of sum of two ideals of an R-group
%J Discussiones Mathematicae. General Algebra and Applications
%D 2023
%P 177-187
%V 43
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMGAA_2023_43_2_a0/
%G en
%F DMGAA_2023_43_2_a0
Sahoo, Tapatee; Kedukodi, Babushri Srinivas; Harikrishnan, Panackal; Kuncham, Syam Prasad. On the finite Goldie dimension of sum of two ideals of an R-group. Discussiones Mathematicae. General Algebra and Applications, Tome 43 (2023) no. 2, pp. 177-187. http://geodesic.mathdoc.fr/item/DMGAA_2023_43_2_a0/

[1] Aichinger E.,Binder F., Ecker F., Mayr P., & Nöbauer C., SONATA - system of near-rings and their applications, GAP package, Version 2.8; 2015. http://www.algebra.uni-linz.ac.at/Sonata/

[2] Anderson Frank W, Fuller, Kent R, Rings and categories of modules, Graduate Texts in Mathematics, Springer-Berlag New York, 13, 1992. https://doi.org/10.1007/978-1-4612-4418-9

[3] Bhavanari S., On modules with finite Goldie Dimension, J. Ramanujan Math. Soc. 5(1), 61-75, 1990.

[4] Bhavanari S., and Kuncham S.P., Linearly independent elements in $ N$-groups with Finite Goldie Dimension, Bull. Korean Math. Soc., 42(3), 433-441, 2005. https://doi.org/10.4134/BKMS.2005.42.3.433

[5] Bhavanari S., Goldie dimension and spanning dimension in modules and $ N $-groups, Nearrings, Nearfields and related topics (Review volume), World Scientific, 26-41, 2017. https://doi.org/10.1142/9789813207363\_0004

[6] Bhavanari S., Contributions to near-ring theory, Doctrol Thesis, Nagarjuna University, India, 1984.

[7] Bhavanari S., and Kuncham S.P., A result on E-direct systems in $ N $-groups, Indian J. pure appl. Math., 29(3), 285-287, 1998.

[8] Bhavanari S., and Kuncham S.P., Nearrings, fuzzy ideals, and graph theory. CRC press, 2013. https://doi.org/10.1201/b14934

[9] Bhavanari S., Kuncham S.P., Paruchuri V.R., and Mallikarjuna B., A note on dimensions in $N$-groups, Italian Journal of Pure and Applied Mathematics, 44, 649-657, 2020.

[10] Goldie A.W., The structure of noetherian rings, Lectures on Rings and Modules, vol 246. Springer, Berlin, Heidelberg, 1972. https://doi.org/10.1007/BFb0059567

[11] Kuncham S.P., Contributions to near-ring theory II. Diss. Doctoral Thesis, Nagarjuna University, 2000.

[12] Kuncham S.P, Kedukodi B.S., Harikrishnan P.K., and Bhavanari S. (Editors), Nearrings, Nearfields and Related Topics. World Scientific Publishing Company, 2017. https://doi.org/10.1142/10375

[13] Nayak Hamsa, Kuncham S.P., Kedukodi B.S., $\Theta$$Γ$ $N$-group,\textit{ Matematicki Vesnik}, 70(1), 64-78, 2018.

[14] Oswald A., Near-rings in which every N-subgroup is principal. Proceedings of the London Mathematical Society 3.1, 67-88, 1974. https://doi.org/10.1112/plms/s3-28.1.67

[15] Oswald A., Completely reducible near-rings, Proceedings of the Edinburgh Mathematical Society, 20(3), 187-197, 1977.

[16] Pilz G., Near-Rings: the theory and its applications, 23, North Holland, 1983.

[17] Tapatee S., Kedukodi B.S., Shum K.P., Harikrishnan P.K., Kuncham S.P., On essential elements in a lattice and Goldie analogue theorem, Asian-Eur. J. Math., 2250091: 2021. https://doi.org/10.1142/S1793557122500917

[18] Yenumula V.R., Bhavanari S., A note on $ N $-Groups, Indian J. Pure Appl. Math., 19(9), 842-845, 1988.