Mathematica Bohemica, first online, pp. 1-10


Direct summands of Goldie extending elements in modular lattices

Rupal Shroff

Received November 29, 2020.   Published online August 17, 2021.

Abstract:  In this paper some results on direct summands of Goldie extending elements are studied in a modular lattice. An element $a$ of a lattice $L$ with $0$ is said to be a Goldie extending element if and only if for every $b \leq a$ there exists a direct summand $c$ of $a$ such that $b \wedge c$ is essential in both $b$ and $c$. Some characterizations of decomposition of a Goldie extending element in a modular lattice are obtained.
Keywords:  modular lattice; direct summand; Goldie extending element
Classification MSC:  06B10, 06C05
DOI:  10.21136/MB.2021.0181-20

PDF available at:  Institute of Mathematics CAS

References:
[1] E. Akalan, G. F. Birkenmeier, A. Tercan: Goldie extending modules. Commun. Algebra 37 (2009), 663-683. DOI 10.1080/00927870802254843 | MR 2493810 | Zbl 1214.16005
[2] G. Călugăreanu: Lattice Concepts of Module Theory. Kluwer Texts in the Mathematical Sciences 22. Kluwer, Dordrecht (2000). DOI 10.1007/978-94-015-9588-9 | MR 1782739 | Zbl 0959.06001
[3] P. Crawley, R. P. Dilworth: Algebraic Theory of Lattices. Prentice Hall, Engelwood Cliffs (1973). Zbl 0494.06001
[4] N. V. Dung, D. V. Huynh, P. F. Smith, R. Wisbauer: Extending Modules. Pitman Research Notes in Mathematics Series 313. Longman Scientific, Harlow (1994). MR 1312366 | Zbl 0841.16001
[5] G. Grätzer: General Lattice Theory. Birkhäuser, Basel (1998). DOI 10.1007/978-3-0348-7633-9 | MR 1670580 | Zbl 0909.06002
[6] P. Grzeszczuk, E. R. Puczyłowski: On Goldie and dual Goldie dimensions. J. Pure Appl. Algebra 31 (1984), 47-54. DOI 10.1016/0022-4049(84)90075-6 | MR 0738204 | Zbl 0528.16010
[7] A. Harmanci, P. F. Smith: Finite direct sums of CS-modules. Houston J. Math. 19 (1993), 523-532. MR 1251607 | Zbl 0802.16006
[8] S. K. Nimbhorkar, R. C. Shroff: Ojective ideals in modular lattices. Czech. Math. J. 65 (2015), 161-178. DOI 10.1007/s10587-015-0166-5 | MR 3336031 | Zbl 1338.06004
[9] S. K. Nimbhorkar, R. C. Shroff: Goldie extending elements in modular lattices. Math. Bohem. 142 (2017), 163-180. DOI 10.21136/MB.2016.0049-14 | MR 3660173 | Zbl 1424.06028
[10] A. Tercan, C. C. Yücel: Module Theory: Extending Modules and Generalizations. Frontiers in Mathematics. Birkhäuser, Basel (2016). DOI 10.1007/978-3-0348-0952-8 | MR 3468915 | Zbl 1368.16002
[11] D. Wu, Y. Wang: Two open questions on Goldie extending modules. Commun. Algebra 40 (2012), 2685-2692. DOI 10.1080/00927872.2011.551902 | MR 2968904 | Zbl 1253.16004

Affiliations:   Rupal Shroff, School of Mathematics and Statistics, MIT World Peace University, S. No. 124, Paud Road, Kothrud, Pune, Maharashtra 411038, India, e-mail: rupal.shroff@mitwpu.edu.in, rupal_shroff84@yahoo.co.in


 
PDF available at: