$d$-$MP$-modules and their localizations

I.O.Melnyk

We investigate the properties of $d$-$MP$-modules, stated using the operator $(\,)_{\#}$. In particular, behavior of prime differential submodules under localizations is determined. We also study the interrelation between quasi-prime and differentially prime submodules of module over an arbitrary associative differential ring. It is established that a differential module is $d$-$MP$-module if every its differentially prime submodule is prime differential.

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2. Gorman Howard E. Differential rings and modules// Scripta Math. -- 1973. -- V.29, №1-2. -- P. 25--35.

3. Nowicki A., Some remarks on d-MP rings// Bull. Acad. pol. sci., Ser. sci. math. -- 1982. -- Vol.30, №7-8. -- P. 311--317.

4. Keigher W. Prime differential ideals in differential rings// Contributions to Algebra, A Collection of Papers Dedicated to Ellis Kolchin / Buss, Cassidy and Kovacic, eds. -- New York: Academic Press -- 1977. -- P. 239--249.

5. Keigher W. F. Quasi-prime ideals in differential rings// Houston J. Math. -- 1978. -- V.4, №3. -- P.~379--388.

6. Kovacic, Jerald J. Differential schemes, Differential algebra and related topics (Newark, NJ, 2000) -- 2002. -- P. 71--94.

7. Kaplansky I. Introduction to differential algebra, Graduate Texts in Mathematics, New York: Springer-Verlag -- 1999. -- V.189. -- 85 p.

8. Kolchin S. E. Differential Algebra and Algebraic Groups. -- New York: Academic Press, 1973. -- 446p.

9. Mikhaliev A. V., Pankratiev E. V. Differential and difference algebra// Itogi nauki i tehniki, Algebra. Topology. Geometry. -- 1987. -- V.25. -- P. 67--134.

10. Khadjiev Dj., Callalp F. On a differential analog of the prime-radical and properties of the lattice of the radical differential ideals in associative differential rings// Tr. J. of Math. -- 1996. -- V.20, №4. -- P.~571--582.

11. Lu~C.~P. Unions of prime submodules// Houston Journal of Mathematics. -- 1997. -- V.23, №2. -- P.~203--213.

12. Lu~C.~P. Spectra of modules// Comm. Algebra. -- 1995. -- V.23. -- P.~3741--3752.

13. Melnyk І. О. $Sdm$-systems, differentially prime and differentially primary modules// Nauk. visnyk Uzhgorod. Univ. Ser. Math. and informat. -- 2008. -- V.16. -- P. 110--118. (Ukrainian)

14. Melnyk І.~О. Differentially prime, quasi--prime and $\Delta$-$MP$-modules// Buletinul academiei de stiinte a republicii Moldova. Matematica -- 2008. -- №3(58). -- P. 112--115.

	$d$-$MP$-modules and their localizations
Author	I.O.Melnyk Algebra and Logic Department, Faculty of Mechanics and Mathematics, Ivan Franko National,University of Lviv, 1 Universytetska St., Lviv, 79000, Ukraine
Abstract	We investigate the properties of $d$-$MP$-modules, stated using the operator $(\,)_{\#}$. In particular, behavior of prime differential submodules under localizations is determined. We also study the interrelation between quasi-prime and differentially prime submodules of module over an arbitrary associative differential ring. It is established that a differential module is $d$-$MP$-module if every its differentially prime submodule is prime differential.
Keywords	d-MP-modules, prime differential submodule, associative differential ring
DOI	doi:10.30970/ms.34.1.13-19
Reference	1. Gorman H. Radical regularity in differential rings// Canad. J. Math. -- 1971. -- V.23, №2. -- P. 197--201. 2. Gorman Howard E. Differential rings and modules// Scripta Math. -- 1973. -- V.29, №1-2. -- P. 25--35. 3. Nowicki A., Some remarks on d-MP rings// Bull. Acad. pol. sci., Ser. sci. math. -- 1982. -- Vol.30, №7-8. -- P. 311--317. 4. Keigher W. Prime differential ideals in differential rings// Contributions to Algebra, A Collection of Papers Dedicated to Ellis Kolchin / Buss, Cassidy and Kovacic, eds. -- New York: Academic Press -- 1977. -- P. 239--249. 5. Keigher W. F. Quasi-prime ideals in differential rings// Houston J. Math. -- 1978. -- V.4, №3. -- P.~379--388. 6. Kovacic, Jerald J. Differential schemes, Differential algebra and related topics (Newark, NJ, 2000) -- 2002. -- P. 71--94. 7. Kaplansky I. Introduction to differential algebra, Graduate Texts in Mathematics, New York: Springer-Verlag -- 1999. -- V.189. -- 85 p. 8. Kolchin S. E. Differential Algebra and Algebraic Groups. -- New York: Academic Press, 1973. -- 446p. 9. Mikhaliev A. V., Pankratiev E. V. Differential and difference algebra// Itogi nauki i tehniki, Algebra. Topology. Geometry. -- 1987. -- V.25. -- P. 67--134. 10. Khadjiev Dj., Callalp F. On a differential analog of the prime-radical and properties of the lattice of the radical differential ideals in associative differential rings// Tr. J. of Math. -- 1996. -- V.20, №4. -- P.~571--582. 11. Lu~C.~P. Unions of prime submodules// Houston Journal of Mathematics. -- 1997. -- V.23, №2. -- P.~203--213. 12. Lu~C.~P. Spectra of modules// Comm. Algebra. -- 1995. -- V.23. -- P.~3741--3752. 13. Melnyk І. О. $Sdm$-systems, differentially prime and differentially primary modules// Nauk. visnyk Uzhgorod. Univ. Ser. Math. and informat. -- 2008. -- V.16. -- P. 110--118. (Ukrainian) 14. Melnyk І.~О. Differentially prime, quasi--prime and $\Delta$-$MP$-modules// Buletinul academiei de stiinte a republicii Moldova. Matematica -- 2008. -- №3(58). -- P. 112--115.
Pages	13-19
Volume	34
Issue	1
Year	2010
Journal	Matematychni Studii
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