ABOUT RING STRUCTURES ON THE SET OF INTEGERS
https://doi.org/10.22405/2226-8383-2017-18-2-6-17
Abstract
It is well known that the ring of integers Z is an ????-ring, therefore it is possible to define unique (up to isomorphism) structure of a ring with identity on the additive group Z. A natural question arises about the uniqueness of the ring structure with identity constructed on a multiplicative monoid Z. It is shown in this paper that this question is solved negatively. Moreover, a method of construction new various ring structures on the multiplicative monoid Z by dint of multiplicative automorphisms was developed and described. The concept of basis was introduced for the multiplicative monoid Z, and it was shown that there are no bases (up to sign) that are differ to a basis consists of all prime numbers, and bases that are obtain of that basis by a permutations of its elements. The example of construction a new ring structure on the set Z for fixed standart multiplication is given in the end of this paper. The new addition on the multiplicative monoid Z is obtained by a permutation of prime numbers (it is 2 ↦→ 3 ↦→ 5 ↦→ 2 permutation in the detailed example). From the results obtained in the paper it follows in particular, that the ring Z is not an unique addition ring (UA-ring).
About the Author
D. Yu. Artemov
Moscow Pedagogical State University
Russian Federation
Student of the Mathematical Department
Russian Federation
Student of the Mathematical Department
References
1. Fuchs, L. 1970, "Infinite Abelian Groups vol. 1, New York–London: Academic Press.
2. Fuchs, L. 1973, "Infinite Abelian Groups vol. 2, New York–London: Academic Press.
3. Kurosh, A. G. 1974, "General Algebra Moscow, Nauka (russian).
4. Schultz, P. 1970, "Periodic homomorphism sequences of abelian groups Arch. Math., vol. 21, pp. 132-135.
5. Schultz, P. 1973, "The endomorphism ring of the additive group of a ring"J. Austral. Math. Soc., vol. 15. pp. 60-69.
6. Bowshell, R. A. & Schultz, P. 1977, "Unital rings whose additive endomorphisms commute Math. Ann., vol. 228, no. 3, pp. 197-214
7. Krylov, P. A., Mikhalev, A. V. & Tuganbaev, A. A. 2013, "Endomorphism rings of Abelian groups vol. 2, Springer Science & Business Media.
8. Feigelstock, S. 1983, "Additive groups of rings Pitman advanced publishing program, London.
9. Arnold, D. M. 1982, "Finite rank torsion free abelian groups and rings Lecture Notes in Math, vol. 931, Springer, New York.
10. Stephenson, W. 1969, "Unique addition rings Canad. J. Math., vol. 21, no. 6, pp. 1455-1461.
11. Nelius, Chr.-F. 1974, "Ring emit eindentiger Addition Padeborn, 1974.
12. Mikhalev, A. V. 1989, "Multiplicative classification of associative rings Mathematics of the USSR-Sbornik, vol. 63, no. 1, pp. 205-218.
13. van der Merwe, B. 1999. "Unique addition modules Communications in algebra, vol. 27, no. 9, pp. 4103–4115.
14. Chistyakov D. S. & Lyubimtsev O. V. 2011, "On abelian torsion free with UA-endomorphism rings Vestnik TGU, no. 2(14), pp. 55–58.
15. Chistyakov D. S. & Lyubimtsev O. V. 2010, "Abelian groups as UA-modules over the ring Z Mathematical Notes, vol. 87, no. 3, pp. 380–383.
Review
For citations:
Artemov D.Yu. ABOUT RING STRUCTURES ON THE SET OF INTEGERS. Chebyshevskii Sbornik. 2017;18(2):6-17. (In Russ.) https://doi.org/10.22405/2226-8383-2017-18-2-6-17
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