Generalization of reduction and closure of ideals

Document Type : Research Paper

Authors

1 Department of Mathematics, Faculty of Science, University of mohaghegh Ardabili, Ardabil

2 Department of Mathematics, University of Mohaghegh Ardabili, Ardabil, Iran

Abstract

Throughout this paper, all rings are commutative  with identity and all modules are unital. Let $R$  be a ring and $M$ be an $R$-module. Then $M$ is called a multiplication module provided for every submodule $N$ of $M$ there exists an ideal $I$ of $R$ such that $N=IM$. Also $M$ is said to be a comultiplication module if for every submodule $N$ of $M$ there exists an ideal $I$ of $R$ such that $ N=(0:_MI)$. In this  paper, we introduce the notions of reduction and coreduction of submodules, integral dependence, integral codependence, integral closure and $\Delta$-closure over multiplication and comultiplication modules.

Keywords

References

[1] Z. Abd El-Bast and P. F. Smith, Multiplication modules , Comm. Algebra. 16(1988), no. 4, 755-779.
[2] M. M. Ali, Idempotent and nilpotent submodules of multiplication modules, for associated primes of local cohomology modules, Comm. Algebra. 36, no. 12, (2008), 4620-4642.
[3] M. M. Ali, Residual submodules of multiplication modules, Beitr. Algebra Geom. 46(2) (2005),405-422.
[4] H. Ansari-Toroghy and F. Farshadifar, Product and dual product of submodules, Far East J. Math. Sci, 25, no. 3, (2007), 447-455.
[5] H. Ansari-Toroghy and F. Farshadifar, The dual notion of multiplication modules, Taiwanese J. Math, 11, no. 4, (2007), 1189-1201.
[6] R. Ameri,on the prime submodules of multiplication modules,The scientific world journal. no.27, (2003), 1715-1724.
[7] A. Barnard, Multiplication modules, Manuscripta Math. 71(1981), no. 1, 174-178.
[8] M.P. Brodmann and R.Y. Sharp, Local cohomology; an algebraic introduction with geometric applications, Cambridge University Press, Cambridge,1998.
[9] H. Matsumura, Commutative ring theory, Cambridge University press, Cambridge, UK, 1986.
[10] R. L. McCasland and M. E. Moore, On radicals of submodules of finitely generated modules, Canad. Math. Bull, 29 (1986) 37-39.
[11] A. G. Naoum and A.S. Mijbass, Weak cancellation modules, Kyungpook Math. J.,37(1997), 73-82.
[12] D. G. Northcott and D. Rees, Reductions of ideals in local rings , Proc. Cambridge Philos. Soc., 50(1954), 145-158.
[13] L. J. Ratliff, Jr., ∆ - closure of ideals and rings, Trans. Amer. Math. Soc., Volume 313, (1989), 221-247.
[14] Y. Tiras, Integral closure of an ideal relative to a module and ∆ - closure , Trans. Amer. Math. Soc.,21(1997), 381-386.
Algebraic Structures and Their Applications
Volume 8, Issue 1
February 2021
Pages 147-161

Files

Share

How to cite

Statistics