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

Access to the full text of journal articles on this site is restricted to the subscribers of Myris Trade. To activate your access, please contact Myris Trade at myris@myris.cz.
Subscribers of Springer need to access the articles on their site, which is http://mb.math.cas.cz/.

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: