On a class of $\lambda$ -modules

I. E.  Wijayanti; M.  Ardiyansyah; P. W. Prasetyo

doi:10.37863/umzh.v73i3.513

I. E. Wijayanti Univ. Gadjah Mada, Indonesia
M. Ardiyansyah Aalto Univ., Finland
P. W. Prasetyo Univ. Ahmad Dahlan, Indonesia

DOI: https://doi.org/10.37863/umzh.v73i3.513

Keywords: lattice of ideals, lattice of submodules, multiplication modules, class of modules

Abstract

UDC 512.5

Smith in paper [{\it Mapping between module lattices}, Int. Electron. J. Algebra, {\bf 15}, 173–195 (2014)] introduced maps between the lattice of ideals of a commutative ring and the lattice of submodules of an $R$-module $M,$ i.e., $\mu$ and $\lambda$ mappings. The definitions of the maps were motivated by the definition of multiplication modules. Moreover, some sufficient conditions for the maps to be a lattice homomorphisms are studied. In this work we define a class of $\lambda$-modules and observe the properties of the class. We give a sufficient conditions for the module and the ring such that the class $\lambda$ is a hereditary pretorsion class.

Author Biography

P. W. Prasetyo, Univ. Ahmad Dahlan, Indonesia

References

M. M. Ali, Invertibility of multiplication modules, New Zealand J. Math., 35, no. 1, 17 – 29 (2006).

M. M. Ali, Invertibility of multiplication modules. II, New Zealand J. Math., 39, 45 – 64 (2009).

M. Alkan, B. Sara¸c, Y. Tira¸s, Dedekind modules, Commun. Algebra, 33, 1617 – 1626 (2005), https://doi.org/10.1081/AGB-200061007 DOI: https://doi.org/10.1081/AGB-200061007

R. Ameri, On the prime submodules of multiplication modules, Int. J. Math. and Math. Sci., 27, no. 27, 1715 – 1724 (2003), https://doi.org/10.1155/S0161171203202180 DOI: https://doi.org/10.1155/S0161171203202180

H. Ansari-Toroghy, F. Farshadifar, On multiplication and comultiplication modules, Acta Math. Sci. Ser. B., 31, № 2, 694 – 700 (2011), https://doi.org/10.1016/S0252-9602(11)60269-5 DOI: https://doi.org/10.1016/S0252-9602(11)60269-5

S. Çeken, M. Alkan, P. F. Smith, The dual notion of the prime radical of a module, J. Algebra, 392, 265 – 275 (2013), https://doi.org/10.1016/j.jalgebra.2013.06.015 DOI: https://doi.org/10.1016/j.jalgebra.2013.06.015

D. D. Anderson, On the ideal equation $I(BC)=IBIC$, Canad. Math. Bull., 26, № 3, 331 – 332 (1983), https://doi.org/10.4153/CMB-1983-054-3

Z. A. El-Bast, P. F. Smith, Multiplication modules, Comm. Algebra, 16, № 4, 755 – 779 (1988), https://doi.org/10.4153/CMB-1983-054-3 DOI: https://doi.org/10.4153/CMB-1983-054-3

M. D. Larsen, P. J. McCarthy, Multiplicative theory of ideals, Acad. Press, Inc., USA (1971).

S. R. Lopez-Permouth, J. E. Simental, Characterizing rings in terms of the extent of the injectivity and projectivity of their modules, J. Algebra, 362, 56 – 69 (2012), https://doi.org/10.1016/j.jalgebra.2012.04.005 DOI: https://doi.org/10.1016/j.jalgebra.2012.04.005

B. Sara¸c, P. F. Smith, Y. Tira¸s, On dedekind modules, Commun. Algebra, 35, no. 5, 1533 – 1538 (2007), https://doi.org/10.1080/00927870601169051 DOI: https://doi.org/10.1080/00927870601169051

P. F. Smith, Mapping between module lattices, Int. Electron. J. Algebra, 15, 173 – 195 (2014), https://doi.org/10.24330/ieja.266246 DOI: https://doi.org/10.24330/ieja.266246

U. Tekir, On multiplication modules, Int. Math. Forum, 2, no. 29-32, 1415 – 1420 (2007),https://doi.org/10.12988/imf.2007.07128 DOI: https://doi.org/10.12988/imf.2007.07128

I. E. Wijayanti, R. Wisbauer, Coprime modules and comodules, Comm. Algebra, 37, № 4, 1308 – 1333 (2009), https://doi.org/10.1080/00927870802466926 DOI: https://doi.org/10.1080/00927870802466926

R. Wisbauer, Grundlagen der Modul- und Ringtheorie: ein Handbuch f¨ur Studium und Forschung, Verlag R. Fischer, M¨unchen (1988).

S. Yassemi, The dual notion of prime submodules, Arch. Math. (Brno), 37, no. 4, 273 – 278 (2001).