Mathematica Bohemica, Vol. 142, No. 2, pp. 163-180, 2017


Goldie extending elements in modular lattices

Shriram K. Nimbhorkar, Rupal C. Shroff

Received June 28, 2014.  First published December 2, 2016.

Abstract:  The concept of a Goldie extending module is generalized to a Goldie extending element in a 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 properties of such elements are obtained in the context of modular lattices. We give a necessary condition for the direct sum of Goldie extending elements to be Goldie extending. Some characterizations of a decomposition of a Goldie extending element in such a lattice are given. The concepts of an $a$-injective and an $a$-ejective element are introduced in a lattice and their properties related to extending elements are discussed.
Keywords:  modular lattice; Goldie extending element
Classification MSC:  06B10, 06C05
DOI:  10.21136/MB.2016.0049-14


References:
[1] E. Akalan, G. F. Birkenmeier, A. Tercan: Corrigendum to: Goldie extending modules. Commun. Algebra 41 (2013), page 2005. Original article ibid. 37 (2009), 663-683 and first correction in 38 (2010), 4747-4748. DOI 10.1080/00927872.2011.651766 | MR 3062842 | Zbl 1278.16005
[2] G. Călugăreanu: Lattice Concepts of Module Theory. Kluwer Texts in the Mathematical Sciences 22. Kluwer Academic Publishers, Dordrecht (2000). DOI 10.1007/978-94-015-9588-9 | MR 1782739 | Zbl 0959.06001
[3] A. W. Chatters, C. R. Hajarnavis: Rings in which every complement right ideal is a direct summand. Q. J. Math., Oxf. (2) 28 (1977), 61-80. DOI 10.1093/qmath/28.1.61 | MR 0437595 | Zbl 0342.16023
[4] P. Crawley, R. P. Dilworth: Algebraic Theory of Lattices. Prentice-Hall, Englewood Cliffs, New Jersey (1973). Zbl 0494.06001
[5] N. V. Dung, D. V. Huynh, P. F. Smith, R. Wisbauer: Extending Modules. Pitman Research Notes in Mathematics Series 313. Longman Scientific & Technical, Harlow (1994). MR 1312366 | Zbl 0841.16001
[6] N. Er: Rings whose modules are direct sums of extending modules. Proc. Am. Math. Soc. 137 (2009), 2265-2271. DOI 10.1090/S0002-9939-09-09807-4 | MR 2495259 | Zbl 1176.16006
[7] G. Grätzer: General Lattice Theory. Birkhäuser, Basel (1998). MR 1670580 | Zbl 0909.06002
[8] 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
[9] A. Harmanci, P. F. Smith: Finite direct sums of CS-modules. Houston J. Math. 19 (1993), 523-532. MR 1251607 | Zbl 0802.16006
[10] M. A. Kamal, A. Sayed: On generalized extending modules. Acta Math. Univ. Comen., New Ser. 76 (2007), 193-200. MR 2385032 | Zbl 1156.16005
[11] D. Keskin: An approach to extending and lifting modules by modular lattices. Indian J. Pure Appl. Math. 33 (2002), 81-86. MR 1879786 | Zbl 0998.16004
[12] T. Y. Lam: Lectures on Modules and Rings. Graduate Texts in Mathematics 189. Springer, New York (1999). DOI 10.1007/978-1-4612-0525-8 | MR 1653294 | Zbl 0911.16001
[13] G. Szász: Introduction to Lattice Theory. Academic Press, New York; Akadémiai Kiadó, Budapest (1963). MR 0166118 | Zbl 0126.03703

Affiliations:   Shriram Khanderao Nimbhorkar, Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad 431004, Maharashtra, India, e-mail: sknimbhorkar@gmail.com; Rupal Chandulal Shroff, Department of Core Engineering and Engineering Sciences, MIT College of Engineering, MIT College Road, Pune 411038, Maharashtra, India, e-mail: rupal_shroff84@yahoo.co.in

 
PDF available at: