# zbMATH — the first resource for mathematics

Some results on CLESS lattices. (English) Zbl 1454.06007
Summary: We introduce the concept of a CLESS lattice which is a generalization of the concept of an extending lattice (or a CS lattice). We study relationship between various generalizations of the concept of an extending lattice namely, CLS lattice, CLESS lattice and CESS lattice. We also prove that, if $$a$$, $$b$$ are direct summands of 1 which are CLESS elements then 1 is a CLESS element.
##### MSC:
 06C20 Complemented modular lattices, continuous geometries 06C05 Modular lattices, Desarguesian lattices
##### Keywords:
modular lattice; CS lattice; CLS lattice; CESS lattice; CLESS lattice
Full Text:
##### References:
 [1] E. Akalan, G. F. Birkenmeier, and A. Tercan, Goldie extending modules,Comm. Algebra,37(2009), 663-683. · Zbl 1214.16005 [2] G. C˘alug˘areanu,Lattice Concepts of Module Theory, Kluwer, Dordrecht,2000. [3] E. Celik, A. Harmanci and P. F. Smith, A generalization of CS-modules,Comm. Algebra,23(1995), 5445-5460. · Zbl 0842.16003 [4] A. W. Chatters and C. R. Hajarnavis, Rings in which every complement right ideal is a direct summand, Quart. J. Math. Oxford(2)28 (1977), 61-80. · Zbl 0342.16023 [5] S. Crivei and S. Sahinkaya, Modules whose closed submodules with essential socle are direct summands, Taiwanese J. Math.,18, no. 4,(2014), 989-1002. · Zbl 1357.16011 [6] N. V. Dung, D. V. Huynh, P. F. Smith and R. Wisbauer,Extending modules, Research Notes in Math. Ser. 313, Pitman, London, 1994. · Zbl 0841.16001 [7] G. Grätzer, Lattice theory:First concepts and distributive lattices, W. H. Freeman and company San Francisco, 1971. · Zbl 0232.06001 [8] P. Grzeszczuk and E. R. Puczylowski, On Goldie and dual Goldie dimension,J. Pure and Appl. Algebra, 31(1984), 47-54. · Zbl 0528.16010 [9] A. Harmanci and P. F. Smith, Finite direct sums of CS-modules,Houston J. Math.,19(1993), 523-532. · Zbl 0802.16006 [10] D. Keskin, An approach to extending and lifting modules by modular lattice,Indian J. Pure Appl. Math. 33, no. 1, (2002), 81-86, · Zbl 0998.16004 [11] T. Y. Lam,Lectures on rings and modules, Springer-Verlag, New York, 1999. · Zbl 0911.16001 [12] B. J. Müller and S. T. Rizvi, Direct sum of indecomposible modules,Osaka J. Math.,21(1984), 365-374. · Zbl 0538.16015 [13] S. K. Nimbhorkar and D. B. Banswal, A note on CESS-lattices, [14] S. K. Nimbhorkar and Rupal Shroff, Ojective ideals in modular lattices,Czech. Math. J.,65(140)(2015), 161-178. · Zbl 1338.06004 [15] S. K. Nimbhorkar and Rupal Shroff, Generalized extending ideals in modular lattices,J. Indian Math. Soc.,82(3-4), (2015), 127-146. · Zbl 1351.06004 [16] S. K. Nimbhorkar and Rupal Shroff, Goldie extending elements in modular lattices,Math. Bohemica142 (2)(2017), 163-180. · Zbl 1424.06028 [17] P. F. Smith,CS-modules and weak CS-modulesNon-commutative Ring Theory Springer LNM 1448 (1990), 99-115. [18] G. Sz¯asz,Introduction to lattice theory, Academic Press, New York, 1963. [19] A. Tercan, On CLS-modules,Rocky Mountain J. Math.,25, no. 4, (1995), 1557 - 1564. · Zbl 0848.16006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.