Geodesic
Goldie extending elements in modular lattices
Mathematica Bohemica, Tome 142 (2017) no. 2, pp. 163-180
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
DOI : 10.21136/MB.2016.0049-14
Classification : 06B10, 06C05
Keywords: modular lattice; Goldie extending element 
@article{10_21136_MB_2016_0049_14,
     author = {Nimbhorkar, Shriram K. and Shroff, Rupal C.},
     title = {Goldie extending elements in modular lattices},
     journal = {Mathematica Bohemica},
     pages = {163--180},
     year = {2017},
     volume = {142},
     number = {2},
     doi = {10.21136/MB.2016.0049-14},
     mrnumber = {3660173},
     zbl = {06738577},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/MB.2016.0049-14/}
}
TY  - JOUR
AU  - Nimbhorkar, Shriram K.
AU  - Shroff, Rupal C.
TI  - Goldie extending elements in modular lattices
JO  - Mathematica Bohemica
PY  - 2017
SP  - 163
EP  - 180
VL  - 142
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.21136/MB.2016.0049-14/
DO  - 10.21136/MB.2016.0049-14
LA  - en
ID  - 10_21136_MB_2016_0049_14
ER  - 
%0 Journal Article
%A Nimbhorkar, Shriram K.
%A Shroff, Rupal C.
%T Goldie extending elements in modular lattices
%J Mathematica Bohemica
%D 2017
%P 163-180
%V 142
%N 2
%U http://geodesic.mathdoc.fr/articles/10.21136/MB.2016.0049-14/
%R 10.21136/MB.2016.0049-14
%G en
%F 10_21136_MB_2016_0049_14
Nimbhorkar, Shriram K.; Shroff, Rupal C. Goldie extending elements in modular lattices. Mathematica Bohemica, Tome 142 (2017) no. 2, pp. 163-180. doi: 10.21136/MB.2016.0049-14

[1] Akalan, E., Birkenmeier, G. F., Tercan, A.: Corrigendum to: Goldie extending modules. Commun. Algebra 41 (2013), page 2005. Original article ibid. {\it 37} (2009), 663-683 and first correction in {\it 38} (2010), 4747-4748. | DOI | MR | JFM

[2] Călugăreanu, G.: Lattice Concepts of Module Theory. Kluwer Texts in the Mathematical Sciences 22. Kluwer Academic Publishers, Dordrecht (2000). | DOI | MR | JFM

[3] Chatters, A. W., Hajarnavis, C. R.: Rings in which every complement right ideal is a direct summand. Q. J. Math., Oxf. (2) 28 (1977), 61-80. | DOI | MR | JFM

[4] Crawley, P., Dilworth, R. P.: Algebraic Theory of Lattices. Prentice-Hall, Englewood Cliffs, New Jersey (1973). | JFM

[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). | MR | JFM

[6] Er, N.: Rings whose modules are direct sums of extending modules. Proc. Am. Math. Soc. 137 (2009), 2265-2271. | DOI | MR | JFM

[7] Grätzer, G.: General Lattice Theory. Birkhäuser, Basel (1998). | MR | JFM

[8] Grzeszczuk, P., Puczyłowski, E. R.: On Goldie and dual Goldie dimensions. J. Pure Appl. Algebra 31 (1984), 47-54. | DOI | MR | JFM

[9] Harmanci, A., Smith, P. F.: Finite direct sums of CS-modules. Houston J. Math. 19 (1993), 523-532. | MR | JFM

[10] Kamal, M. A., Sayed, A.: On generalized extending modules. Acta Math. Univ. Comen., New Ser. 76 (2007), 193-200. | MR | JFM

[11] Keskin, D.: An approach to extending and lifting modules by modular lattices. Indian J. Pure Appl. Math. 33 (2002), 81-86. | MR | JFM

[12] Lam, T. Y.: Lectures on Modules and Rings. Graduate Texts in Mathematics 189. Springer, New York (1999). | DOI | MR | JFM

[13] Szász, G.: Introduction to Lattice Theory. Academic Press, New York; Akadémiai Kiadó, Budapest (1963). | MR | JFM

Cité par Sources :