The endomorphisms algebra of translations group and associative unitary ring of trace-preserving endomorphisms in affine plane




Affine plane, Endomorphisms, Trace-preserving endomorphisms, Translation group, Aditive group, Associative ring


A description of Endomorphisms of the translation group is introduced in an affine plane, will define the addition and composition of the set of endomorphisms and specify the neutral elements associated with these two actions and present the Endomorphism algebra thereof will distinguish the Trace-preserving endomorphism algebra in affine plane, and prove that the set of Trace-preserving endomorphism associated with the ’addition’ action forms a commutative group. We also try to prove that the set of trace-preserving endomorphism, together with the two actions, in it, ’addition’ and ’composition’ forms an associative and unitary ring.

Author Biographies

Orgest Zaka, Agricultural University of Tirana.

Dept. of Mathematics-Informatics.

Mohanad. A. Mohammed, Open Educational College.

Dept. of Mathematics.


E. Artin, Geometric algebra. New York, NY: Wiley Interscience, 1988, doi: 10.1002/9781118164518

M. Berger, Geometry, vol. 1. Berlin: Springer, 1987, doi: 10.1007/978-3-540-93815-6.

M. Berger, Geometry, vol. 2. Berlin: Springer, 1987, doi: 10.1007/978-3-540-93816-3

H. S. M. Coxeter, Introduction to geometry, 2nd ed. New York, NY: Wiley, 1989.

K. Filipi, O. Zaka, and A. Jusufi, “The construction of a corp in the set of points in a line of desargues affine plane”, Matemati?ki bilten, vol. 43, no. 1, pp., pp. 27-46, 2019, doi: 10.37560/matbil1190027k

R. C. Hartshorne, Foundations of projective geometry. New York, NY: Benjamin/Cummings, 1967.

D. Hilbert, The foundations of geometry, La Salle, IL: The Open Court Publishing, 1959.

D. R. Hughes and F.C. Piper, Projective planes. Berlin: Spriger, 1973

A. Kryftis, “Aconstructive approach to affine and projective planes”, Jan. 2016. arXiv: 1601.04998v1

S. Lang, Algebra, 3rd ed. New York, NY: Springer, doi: 10.1007/978-1-4613-0041-0

J. J. Rotman, Advanced modern algebra, 2nd ed. Providence, RI: American mathematical society., 2010.

R. Wisbauer, Foundations of module and ring theory. Philadelphia, PA: Gordon and Breach Science Publishers, 1991.

O. Zaka, "Contribution to reports of some algebraic structures with affine plane geometry and applications", Ph. D thesis, Polytechnic University of Tirana, Department of Mathematical Engineering, 2016, doi: 10.13140/RG.2.2.13705.08804

O. Zaka, “Three vertex and parallelograms in the affine plane: similarity and addition abelian groups of similarly n-vertexes in the Desargues affine plane”, Mathematical modelling and applications, vol. 3, no. 1, pp. 9-15, 2018, doi: 10.11648/

O. Zaka, “Dilations of line in itself as the automorphism of the skew-field constructed over in the same line in Desargues affine plane”, Applied mathematical sciences, vol. 13, no. 5, pp. 231–237, 2019, doi: 10.12988/ams.2019.9234

O. Zaka and K. Filipi, “The transform of a line of Desargues affine plane in an additive group of its points”, International journal. of current research, vol. 8, no. 7, pp. 34983-34990, Jul. 2016. [On line]. Available:

O. Zaka and J.F. Peters, “Isomorphic-dilations of the skew-fields constructed over parallel lines in the desargues affine plane”, Mar. 2019, arXiv:1904.01469v1

O. Zaka and J.F. Peters, “Ordered line and skew-fields in the Desargues affine plane”, May 2019, arXiv:1905.03859

O. Zaka, “One construction of an affine plane over a corps”, Journal of advances in mathematics, vol. 12, no. 5, pp. 6200-6206, 2016,

O. Zaka, “A description of collineations-groups of an affine plane”, Libertas mathematica, vol. 37, no. 2, pp. 81-96, 2017. [On line]. Available:



How to Cite

O. Zaka and M. A. Mohammed, “The endomorphisms algebra of translations group and associative unitary ring of trace-preserving endomorphisms in affine plane”, Proyecciones (Antofagasta, On line), vol. 39, no. 4, pp. 821-834, Jul. 2020.