[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.4134/CKMS.c160103

ON SUBDIRECT PRODUCT OF PRIME MODULES

Dehghani, Najmeh (Department of Mathematics Persian Gulf University)
Vedadi, Mohammad Reza (Department of Mathematical Sciences Isfahan University of Technology)

Publication Information

Communications of the Korean Mathematical Society / v.32, no.2, 2017 , pp. 277-285 More about this Journal

Abstract

In the various module generalizations of the concepts of prime (semiprime) for a ring, the question "when are semiprime modules subdirect product of primes?" is a serious question in this context and it is considered by earlier authors in the literature. We continue study on the above question by showing that: If R is Morita equivalent to a right pre-duo ring (e.g., if R is commutative) then weakly compressible R-modules are precisely subdirect products of prime R-modules if and only if dim(R) = 0 and R/N(R) is a semi-Artinian ring if and only if every classical semiprime module is semiprime. In this case, the class of weakly compressible R-modules is an enveloping for Mod-R. Some related conditions are also investigated.

Keywords

classical prime module; prime module; semi-Artinian ring; semiprime module; weakly compressible module;

Citations & Related Records

Reference

1	S. M. George, R. L. McCasland, and P. F. Smith, A principal ideal theorem analougue for modules over commutative rings, Comm. Algebra 22 (1994), no. 6, 2083-2099. DOI
2	A. Haghany and M. R. Vedadi, Endoprime modules, Acta Math. Hungar. 106 (2005), no. 1-2, 89-99. DOI
3	L. Levy, Torsion-free and divisible modules over non-integral domains, Canad. J. Math. 15 (1963), 132-151. DOI
4	C. Lomp, Prime elements in partially ordered groupoids applied to modules and Hopf algebra actions, J. Algebra Appl. 4 (2005), no. 1, 77-97. DOI
5	J. C. McConnell and J. C. Robson, Non-commutative Noetherian Rings, Wiley-Interscience, New York, 1987.
6	P. F. Smith and M. R. Vedadi, Essentially compressible modules and rings, J. Algebra 304 (2006), no. 2, 812-831. DOI
7	P. F. Smith and M. R. Vedadi, Submodules of direct sums of compressible modules, Comm. Algebra 36 (2008), no. 8, 3042-3049. DOI
8	Y. Tolooei and M. R. Vedadi, On rings whose modules have nonzero homomorphisms to nonzero submodules, Publ. Mat. 57 (2013), no. 1, 107-122. DOI
9	R. Wisbauer, Modules and Algebras: Bimodule Structure and Group Action on Algebras, Pitman Monograhs 81, Addison-Wesley-Longman, 1996.
10	H. P. Yu, On quasiduo rings, Glasgow Math. J. 37 (1995), no. 1, 21-31. DOI
11	J. Zelmanowitz, Weakly semisimple modules and density theory, Comm. Algebra 21 (1993), no. 5, 1785-1808. DOI
12	E. Enochs, O. M. G. Jenda, and J. Xu, The existence of envelopes, Rend. Sem. Mat. Univ. Padova 90 (1993), 45-51.
13	F. W. Anderson and K. R. Fuller, Rings and categories of modules, Granduate Text in Mathematics 13, Springer, Berline, 1973.
14	O. D. Avraamova, A generalized density theorem, in Abelian groups and modules, No. 8 (Russian), 3-16, 172, Tomsk. Gos. Univ., Tomsk, 1989.
15	M. Behboodi, A generalization of Baer's lower nilradical for modules, J. Algebra Appl. 6 (2007), no. 2, 337-353. DOI
16	L. Bican, P. Jambor, T. Kepka, and P. Nemec, Prime and coprime modules, Fund. Math. 57 (1980), no. 1, 33-45.
17	N. Dehghani and M. R. Vedadi, A characterization of modules embedding in products of primes and enveloping condition for their class, J. Algebra Appl. 14 (2015), no. 4, 1550051, 14 pp.