DOI: 10.12732/ijam.v38i3.6
PARTIAL ORDER IN MODULE OVER
A MATRIX NEARRING
S. Tapatee 1, B.S. Kedukodi 2, P. Pallavi 3, P.K. Harikrishnan 4, S.P. Kuncham 5,ยง
1,3 Department of Mathematics
Manipal Institute of Technology Bengaluru
Manipal Academy of Higher Education, INDIA
2,4,5 Department of Mathematics
Manipal Institute of Technology
Manipal Academy of Higher Education
Manipal, Karnataka, INDIA
Abstract. Let $M_n(N)$ be a matrix nearring over the nearring N with identity and let $N^n$ be the direct sum of n-copies of the group (N, +). We introduce a partial order in the $M_n(N)$-group $N^n$ corresponding to the partial order in N-group (over itself). We define a positive cone in $M_n(N)$-group $N^n$ and obtain its characterization. For a convex ideal of $_{N}{N}$, the corresponding ideal in $M_n(N )$-group $N^n$ is described; and conversely, if $I$ is a convex ideal in $M_n(N)$-group $N^n$, then the ideal $I**$ is convex in N (over itself). This establishes the one-one correspondence between the convex ideals of the p.o. N-group $_{N}{N}$ and those of p.o. $M_n(N)$-group $N^{n}$.
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DOI: 10.12732/ijam.v38i3.6
Source: International Journal of Applied Mathematics
ISSN printed version: 1311-1728
ISSN on-line version: 1314-8060
Year: 2025
Volume: 38
Issue: 3
References
[1] G.L. Booth, N.J. Groenewald, On primeness in matrix near-rings, Arch. Math., 56 (1991), 539-546.
[2] S. Bhavanari, S.P. Kuncham, Near Rings, Fuzzy Ideals, and Graph Theory, CRC Press, New York (2013).
[3] S. Bhavanari, M.B.V. Lokeswara Rao, and S.P. Kuncham, A note on primeness in near-rings and matrix near-rings, Indian J. Pure Appl. Math., 27 (1996), 227-234.
[4] S. Bhavanari, S.P. Kuncham, On finite Goldie dimension of Mn(N)-group Nn, In: Nearrings and Nearfields: Proc. of the Conference on Nearrings and Nearfields, Hamburg, Germany, Springer, Netherlands (2005), 301-310.
[5] L. Fuchs, Partially Ordered Algebraic Systems, Courier Corporation, 28 (2011).
[6] M. Jingjing, Lecture Notes on Algebraic Structure of Lattice-ordered Rings, World Scientific (2014).
[7] S.P. Kuncham, et al., Nearrings, Nearfields and Related Topics, World Scientific (2016).
[8] G. Pilz, Direct sums of ordered near-rings, J. Algebra, 18, No 2 (1971), 340-342.
[9] G. Pilz, Near-rings: The Theory and its Applications, Elsevier (2011).
[10] J.D.P. Meldrum, John, A. P. J. Van der Walt, Matrix near-rings, Arch. Math., 47, (1986), 312-319.
[11] J.D.P. Meldrum, J.H. Meyer, Intermediate ideals in matrix nearrings, Commun. Algebra, 24, No 5 (1996), 1601-1619.
[12] J.H. Meyer, Left ideals in matrix near-rings, Commun. Algebra, 17, No 6 (1989), 1315-1335.
[13] A. Radhakrishna, A., M.C. Bhandari, On a class of lattice ordered nearrings, Indian J. Pure Appl. Math, 9, No 6 (1977), 581-587.
[14] A. P. J. Van der Walt, Primitivity in matrix near-rings, Quaestiones Math., 9, No (1-4) (1986), 459-469.
[15] S. Tapatee, B.S. Kedukodi, S. Juglal, P.K. Harikrishnan and S.P. Kuncham, Generalization of prime ideals in Mn(N)-group Nn, Rend. Circ. Mat. Palermo Series 2, 72, No 1 (2023), 449-465.
[16] S. Tapatee, B. Davvaz, P.K. Harikrishnan, B.S. Kedukodi and S.P. Kuncham, Relative essential ideals in N-groups, Tamkang J. Math., 54, No 1 (2023), 69-82.
[17] S. Tapatee, J.H. Meyer, P.K. Harikrishnan, B.S. Kedukodi and S.P. Kuncham, Partial order in matrix nearrings, Bull. Iran. Math. Soc., 48, No 6 (2022), 3195-3209.
[18] S. Rajani, S. Tapatee, P.K. Harikrishnan, B.S. Kedukodi and S.P. Kuncham, Superfluous ideals of N-groups, Rend. Circ. Mat. Palermo Series 2, 72, No 8 (2023), 4149-4167.
[19] K.B. Prabhakara, Extensions of strict partial orders in N-groups, J. Aust. Math. Soc., 25, No 2 (1978), 241-249
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