Some relationships between different types of spectrums of multiplicational
modules and with spectral spaces

M. O. Maloid-Glebova

In this paper we introduce and study the left spectrum and the torsion-theoretic spectrum of modules and establish some connections between them. In particular, the fact that for a multiplication module the left torsion-theoretic spectrum is a quotient space is proved. Also, the notion of locally-prime torsion theory is given, some properties of such torsion theories are presented. Is proved the fact, that symmetric extended (by Belluce) torsion-theoretic spectrum of a ring (module), under certain restrictions, is a spectral space. Also we construct spectrum for the left spectrum of a ring, that, in its turn, is based on the theory of strongly-prime modules. As a technical equipment we use general facts from the torsion theory in the category of left modules over associative rings and some generalizations of technical results from previously published work of many authors.

1. Andrunakievich V. Prime modules and Baer radical// Siberian Mathematical Journal. – 1961. – V.2, №6. – P. 801–806. (in Russian)

2. Andrunakievich V.A., Riabuhin Yu.M. Special modules and special radicals// DAN SSSR. – 1962. – V.147, №6. – P. 1274–1277. (in Russian)

3. Maloid-Glebova M.O. On torsion-theoretic spectrum of left-invariant ring and weakly-multiplication and pure-multiplicayion modules// Applied Problems of Mechanics and Mathematics. – 2011. – V.9. – P. 87– 94. (in Ukrainian)

4. Azizi A. Radical formula and prime submodules// J. Algebra. – 2007. – V.307. – P. 454–460.

5. Bican L., Jampor P., Kepka T., Nemec P. Prime and coprime modules// Fund. Math. – 1980. – V.57. – P. 33–45.

6. Beachy J.A. Some aspects of noncommutative localization// In book: Noncommutative Ring Theor, LNM, Spriger-Verlag, Berlin. – 1975. – V.545. – P. 2–31.

7. Behboodi M., Koohy H. Weakly prime modules// Vietnam J. Math. – 2004. – V.32, №2. – P. 185–195.

8. Belluce L.P. Spectral closure for non-commutative rings// Communications in Algebra. – 1997. – V.25, №5. – P. 1513–1536.

9. Dauns J. Prime modules// J. Reine Angew. Math. – 1978. – V.298. – P. 156–181.

10. Golan J.S. Topologies on the torsion-theoretic spectrum of a noncommutatie ring// Pacific Journal of Mathematics. – 1974. – V.51, №2. – P. 439–450.

11. Golan J.S., Torsion theories. – Longman Scientific & Technical, Harlow, 1986, 651 p.

12. Feller E.H., Swokowski E.W. Prime modules// Can. J. Math. – 1965. – V.17. – P. 1041–1052.

13. Handelman D., Lawrence J. Strongly prime rings// Trans. Amer. Math. Soc. – 1975. – V.211. – P. 209– 223.

14. Hochster M. Prime ideal structure in commutative rings// Trans. Amer. Math. Soc. – 1969. – V.137. – P. 43–60.

15. Jara P., Verhaege P., Verschoren A. On the left spectrum of a ring// Comm. Algebra. – 1994. – V.22, №8. – P. 2983–3002.

16. Johnson R.E. Representations of prime rings// Trans. Amer. Math. Soc. – 1953. – V.74, №2. – P. 351–357.

17. Karakas H.I. On Noetherian modules// Journal of Pure and Applied Science. – 1972. – P. 165–168.

18. Kaucikas A. On the left strongly prime modules and ideals// Liet. Mat. Rink, Special Issue. – 2001. – V.41. – P. 84–87.

19. Kaucikas A. On the left strongly prime modules and their radicals// Lietuvos matematikos rinkinys. – 2010. – V.51. – P. 31–34.

20. Kaucikas A., Wisbauer R. On strongly prime rings and ideals// Comm. Algebra. – 2000. – V.28, №11. – P. 5461–5473.

21. Klept I., Tressl M. The prime spectrum and extended prime spectrum of noncommutative rings// Algebr Represent Theor. – 2007. – V.10. – P. 257–270.

22. Lambek J., Michler G. The torsion theory at a prime ideal of a right Noetherian ring// J. Algebra. – 1974. – V.25. – P. 364–389.

23. Lu C.P. Spectra of modules// Comm. Algebra. – 1995. – V.23. – P. 3741–3752.

24. McCasland R.L., Moore M.E., Smith P.F. On the spectrum of the module over a commutative ring// Comm. Algebra. – 1997. – V.25, №1. – P. 79–103.

25. McCoy N.H. Rings and ideals. – Carus Mathematical Monographs, Math. Assoc. Amer., Menasha, WI: George Banta, 1962, №8.

26. Page S. Properties of quotient rings// Can. J. Math. – 1972. – V.24, №6. – P. 1122–1128.

27. Popescu N. Abelian cathegories with applications to rings and modules. – Academic press, London-New York, 1973, 357 p.

28. Tuganbaev A.A. Multiplication modules// Journal of Mathematical Sciences. – 2004. – V.123, №2. – P. 3839–3905.

29. Shigenaga K. On some prime modules// Res. Rep. of U-be. Tech. Coll. – 1982. – №28. – P. 1–8.

30. Rosenberg A. Noncommutative algebraic geometry and representations of quantized algebras. – Kluwer Academic Publishers, 1995, 316 p.

31. Wisbauer R. On prime modules and rings// Commun. Algebra. – 1983. – V.11. – P. 2249–2265.

	Some relationships between different types of spectrums of multiplicational modules and with spectral spaces(in Ukrainian)
Author	M. O. Maloid-Glebova martamaloid@gmail.com Ivan Franko National University of Lviv
Abstract	In this paper we introduce and study the left spectrum and the torsion-theoretic spectrum of modules and establish some connections between them. In particular, the fact that for a multiplication module the left torsion-theoretic spectrum is a quotient space is proved. Also, the notion of locally-prime torsion theory is given, some properties of such torsion theories are presented. Is proved the fact, that symmetric extended (by Belluce) torsion-theoretic spectrum of a ring (module), under certain restrictions, is a spectral space. Also we construct spectrum for the left spectrum of a ring, that, in its turn, is based on the theory of strongly-prime modules. As a technical equipment we use general facts from the torsion theory in the category of left modules over associative rings and some generalizations of technical results from previously published work of many authors.
Keywords	module spectrum; multiplication module; locally-prime torsion theory; spectral space
Reference	1. Andrunakievich V. Prime modules and Baer radical// Siberian Mathematical Journal. – 1961. – V.2, №6. – P. 801–806. (in Russian) 2. Andrunakievich V.A., Riabuhin Yu.M. Special modules and special radicals// DAN SSSR. – 1962. – V.147, №6. – P. 1274–1277. (in Russian) 3. Maloid-Glebova M.O. On torsion-theoretic spectrum of left-invariant ring and weakly-multiplication and pure-multiplicayion modules// Applied Problems of Mechanics and Mathematics. – 2011. – V.9. – P. 87– 94. (in Ukrainian) 4. Azizi A. Radical formula and prime submodules// J. Algebra. – 2007. – V.307. – P. 454–460. 5. Bican L., Jampor P., Kepka T., Nemec P. Prime and coprime modules// Fund. Math. – 1980. – V.57. – P. 33–45. 6. Beachy J.A. Some aspects of noncommutative localization// In book: Noncommutative Ring Theor, LNM, Spriger-Verlag, Berlin. – 1975. – V.545. – P. 2–31. 7. Behboodi M., Koohy H. Weakly prime modules// Vietnam J. Math. – 2004. – V.32, №2. – P. 185–195. 8. Belluce L.P. Spectral closure for non-commutative rings// Communications in Algebra. – 1997. – V.25, №5. – P. 1513–1536. 9. Dauns J. Prime modules// J. Reine Angew. Math. – 1978. – V.298. – P. 156–181. 10. Golan J.S. Topologies on the torsion-theoretic spectrum of a noncommutatie ring// Pacific Journal of Mathematics. – 1974. – V.51, №2. – P. 439–450. 11. Golan J.S., Torsion theories. – Longman Scientific & Technical, Harlow, 1986, 651 p. 12. Feller E.H., Swokowski E.W. Prime modules// Can. J. Math. – 1965. – V.17. – P. 1041–1052. 13. Handelman D., Lawrence J. Strongly prime rings// Trans. Amer. Math. Soc. – 1975. – V.211. – P. 209– 223. 14. Hochster M. Prime ideal structure in commutative rings// Trans. Amer. Math. Soc. – 1969. – V.137. – P. 43–60. 15. Jara P., Verhaege P., Verschoren A. On the left spectrum of a ring// Comm. Algebra. – 1994. – V.22, №8. – P. 2983–3002. 16. Johnson R.E. Representations of prime rings// Trans. Amer. Math. Soc. – 1953. – V.74, №2. – P. 351–357. 17. Karakas H.I. On Noetherian modules// Journal of Pure and Applied Science. – 1972. – P. 165–168. 18. Kaucikas A. On the left strongly prime modules and ideals// Liet. Mat. Rink, Special Issue. – 2001. – V.41. – P. 84–87. 19. Kaucikas A. On the left strongly prime modules and their radicals// Lietuvos matematikos rinkinys. – 2010. – V.51. – P. 31–34. 20. Kaucikas A., Wisbauer R. On strongly prime rings and ideals// Comm. Algebra. – 2000. – V.28, №11. – P. 5461–5473. 21. Klept I., Tressl M. The prime spectrum and extended prime spectrum of noncommutative rings// Algebr Represent Theor. – 2007. – V.10. – P. 257–270. 22. Lambek J., Michler G. The torsion theory at a prime ideal of a right Noetherian ring// J. Algebra. – 1974. – V.25. – P. 364–389. 23. Lu C.P. Spectra of modules// Comm. Algebra. – 1995. – V.23. – P. 3741–3752. 24. McCasland R.L., Moore M.E., Smith P.F. On the spectrum of the module over a commutative ring// Comm. Algebra. – 1997. – V.25, №1. – P. 79–103. 25. McCoy N.H. Rings and ideals. – Carus Mathematical Monographs, Math. Assoc. Amer., Menasha, WI: George Banta, 1962, №8. 26. Page S. Properties of quotient rings// Can. J. Math. – 1972. – V.24, №6. – P. 1122–1128. 27. Popescu N. Abelian cathegories with applications to rings and modules. – Academic press, London-New York, 1973, 357 p. 28. Tuganbaev A.A. Multiplication modules// Journal of Mathematical Sciences. – 2004. – V.123, №2. – P. 3839–3905. 29. Shigenaga K. On some prime modules// Res. Rep. of U-be. Tech. Coll. – 1982. – №28. – P. 1–8. 30. Rosenberg A. Noncommutative algebraic geometry and representations of quantized algebras. – Kluwer Academic Publishers, 1995, 316 p. 31. Wisbauer R. On prime modules and rings// Commun. Algebra. – 1983. – V.11. – P. 2249–2265.
Pages	3-17
Volume	41
Issue	1
Year	2014
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Some relationships between different types of spectrums of multiplicational modules and with spectral spaces(in Ukrainian)