Quasi-semiprime comultiplication modules over pullback rings

Authors
Farkhondeh Farzalipour 1
Peyman Ghiasvand 2
1 Payame Noor University
2 Departme Payame Noor University
Abstract
The main purpose of this article is to classify all indecomposable quasi-semiprime comultiplication modules over pullback rings of two Dedekind domains and establish a connection between the quasi-semiprime comultiplication modules and the pure-injective modules over such rings. First, we introduce and study the notion of quasi-semiprime comultiplication modules and classify quasi-semiprime comultiplication modules over local Dedekind domains. Second, we get all indecomposable separated quasi-semiprime comultiplication modules and then, using this list of separated quasi-semiprime comultiplication modules, we show that non-separated indecomposable quasi-semiprime comultiplication Rmodules with finite-dimensional top are factor modules of finite direct sums of separated indecomposable quasi-semiprime comultiplication R-modules.
Keywords
Dedekind domains
Pullback
Pure-injective modules
Separated modules
Quasi-semiprime comultiplication modules
1. Ebrahimi Atani S., On pure-injective modules over pullback rings, Comm. Algebra, 28 (2000) 4037-4069.

2. Ebrahimi Atani S., On secondary modules over Dedekind domains, Southeast Asian Bull. Math., 25(25) (2001) 1-6.

3. Ebrahimi Atani S., On secondary modules over pullback rings, Comm. Algebra, 30 (2002) 2675-2685.

4. Ebrahimi Atani S., Indecomposable weak multiplication modules over Dedekind domains, Demonstratio Math., 41 (2008) 33-43.

5. Ebrahimi Atani S., Dolati Pish Hesari S., Khoramdel M., Sedghi Shanbeh Bazari M., Absorbing comultiplication modules over a pullback ring, Int. Electron. J. Algebra, 24 (2018) 31-49.

6. Ebrahimi Atani R., Ebrahimi Atani S., comultiplication modules over a pullback of Dedekind domains, Czechoslovak Math. J., 59 (2009) 1103-1114.

7. Ebrahimi Atani R., Ebrahimi Atani S., On semiprime comultiplication modules over pullback rings, Colloquium Math., 146(2) (2016) 1-15.

8. Ebrahimi Atani R., Ebrahimi Atani S., Weak comultiplication modules over a pullback of commutative local Dedekind domains, Algebra and Discrete Mathematics, 1 (2009) 1-13.

9. Ebrahimi Atani S., Esmaeili Khalil Saraei F., Indecomposable primary comultiplication modules over a pullback of two Dedekind domains, Colloquium Math., 120 (2010) 23-42.

10. Klingler L., Integral representation of groups of square-free order, J. Algebra, 129 (1990) 26-74.

11. Kaplansky I., Modules over Dedekind rings and valuation rings, Trans. Amer. Math. Soc., (1952) 327-340.

12. Kielpiniki R., On Γ-pure-injective modules, Bull. Acad. Sci. Math., 15 (1967) 127-131.

13. Levy L.S., Modules over pullbacks and subdirect sums, J. Algebra, 71 (1981) 50-61.

14. Levy L.S., Modules over Dedekind-like rings, J. Algebra, 93 (1985) 1-116.

15. Levy L.S., Mixed modules over ZG, G, cyclic of prime order, and over related Dedekind pullback, J. Algebra, 71 (1981) 62-114.

16. McCasland R.L., Moore M.E., Smith P.F., On the spectrum of a module over a commutative ring, Comm. Algebra, 25(1) (1997) 79-103.

17. Prest M., Model Theory and Modules, Cambridge: London Mathematical Society, Cambridge University Press, 1988.

18. Warfield R.B., Purity and algebraic compactness for modules, Pacific J. Math., 28 (1969) 699-719.