Goldie extending elements in modular lattices

Shriram K. Nimbhorkar; Rupal C. Shroff

Mathematica Bohemica (2017)

  • Volume: 142, Issue: 2, page 163-180
  • ISSN: 0862-7959

Abstract

top
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 ≤ a there exists a direct summand c of a such that b ∧ 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.

How to cite

top

Nimbhorkar, Shriram K., and Shroff, Rupal C.. "Goldie extending elements in modular lattices." Mathematica Bohemica 142.2 (2017): 163-180. <http://eudml.org/doc/288109>.

@article{Nimbhorkar2017,
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 $b \le a$ 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.},
author = {Nimbhorkar, Shriram K., Shroff, Rupal C.},
journal = {Mathematica Bohemica},
keywords = {modular lattice; Goldie extending element},
language = {eng},
number = {2},
pages = {163-180},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Goldie extending elements in modular lattices},
url = {http://eudml.org/doc/288109},
volume = {142},
year = {2017},
}

TY - JOUR
AU - Nimbhorkar, Shriram K.
AU - Shroff, Rupal C.
TI - Goldie extending elements in modular lattices
JO - Mathematica Bohemica
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 142
IS - 2
SP - 163
EP - 180
AB - 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 $b \le a$ 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.
LA - eng
KW - modular lattice; Goldie extending element
UR - http://eudml.org/doc/288109
ER -

References

top
  1. Akalan, E., Birkenmeier, G. F., Tercan, A., 10.1080/00927872.2011.651766, Commun. Algebra 41 (2013), page 2005. Original article ibid. (2009), 663-683 and first correction in (2010), 4747-4748. (2013) Zbl1278.16005MR3062842DOI10.1080/00927872.2011.651766
  2. Călugăreanu, G., 10.1007/978-94-015-9588-9, Kluwer Texts in the Mathematical Sciences 22. Kluwer Academic Publishers, Dordrecht (2000). (2000) Zbl0959.06001MR1782739DOI10.1007/978-94-015-9588-9
  3. Chatters, A. W., Hajarnavis, C. R., 10.1093/qmath/28.1.61, Q. J. Math., Oxf. (2) 28 (1977), 61-80. (1977) Zbl0342.16023MR0437595DOI10.1093/qmath/28.1.61
  4. Crawley, P., Dilworth, R. P., Algebraic Theory of Lattices, Prentice-Hall, Englewood Cliffs, New Jersey (1973). (1973) Zbl0494.06001
  5. Dung, N. V., Huynh, D. V., Smith, P. F., Wisbauer, R., Extending Modules, Pitman Research Notes in Mathematics Series 313. Longman Scientific & Technical, Harlow (1994). (1994) Zbl0841.16001MR1312366
  6. Er, N., 10.1090/S0002-9939-09-09807-4, Proc. Am. Math. Soc. 137 (2009), 2265-2271. (2009) Zbl1176.16006MR2495259DOI10.1090/S0002-9939-09-09807-4
  7. Grätzer, G., General Lattice Theory, Birkhäuser, Basel (1998). (1998) Zbl0909.06002MR1670580
  8. Grzeszczuk, P., Puczyłowski, E. R., 10.1016/0022-4049(84)90075-6, J. Pure Appl. Algebra 31 (1984), 47-54. (1984) Zbl0528.16010MR0738204DOI10.1016/0022-4049(84)90075-6
  9. Harmanci, A., Smith, P. F., Finite direct sums of CS-modules, Houston J. Math. 19 (1993), 523-532. (1993) Zbl0802.16006MR1251607
  10. Kamal, M. A., Sayed, A., On generalized extending modules, Acta Math. Univ. Comen., New Ser. 76 (2007), 193-200. (2007) Zbl1156.16005MR2385032
  11. Keskin, D., An approach to extending and lifting modules by modular lattices, Indian J. Pure Appl. Math. 33 (2002), 81-86. (2002) Zbl0998.16004MR1879786
  12. Lam, T. Y., 10.1007/978-1-4612-0525-8, Graduate Texts in Mathematics 189. Springer, New York (1999). (1999) Zbl0911.16001MR1653294DOI10.1007/978-1-4612-0525-8
  13. Szász, G., Introduction to Lattice Theory, Academic Press, New York; Akadémiai Kiadó, Budapest (1963). (1963) Zbl0126.03703MR0166118

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.