Extending property on EC-Fully Submodules

Yıl 2018, Cilt: 22 Sayı: 6, 1799 - 1803, 01.12.2018

Ramazan Yaşar Adnan Tercan

https://doi.org/10.16984/saufenbilder.420612

Öz

There
are several generalizations of -modules
in literature. One of the generalization is based on fully invariant
submodules. Recall that a
module  is called -extending
if every fully invariant submodule is essential in a direct summand. We call a module
-extending
if every fully invariant submodule which contains essentially a cyclic
submodule is essential in a direct summand. Initially we obtain basic
properties in the general module setting. For example, a direct sum of -extending
modules is -extending.
Again, like the -extending
property, the -extending
property is shown to carry over to matrix rings.

Anahtar Kelimeler

fully invariant, ec-fully submodule, FI-extending, extending, extending

Kaynakça

• [1] Birkenmeier, G. F., Müller, B. J. and Rizvi S. T., “Modules in which every fully invariant submodule is Essential in a Direct Summand,” Communications in Algebra, vol. 30, no.3, pp. 1395-1415, 2007. [2] Birkenmeier, G. F., Park, J. K. and Rizvi, S. T., Extensions of rings and modules, New York: Birkhäuser, 2013.[3] Celep, C. and Tercan A., "Modules whose ec-closed submodules are direct summand," Taiwanese Journal of Mathematics, vol.13, no.4, pp. 1247-1256, 2009.[4] Dung, N. V., Huynh, D. V., Smith, P. F., and Wisbauer, R., Extending Modules, Pitman Research Notes in Mathematics Series, 313, Longmon, New York, 1994.[5] Kamal, M. A., and Elmnophy, O. A., "On P-extending modules," Acta Math. Univ. Comenianae, vol. 74, no. 2, pp. 279-286, 2005.[6] Kara, Y., Tercan, A. and Yaşar R., "PI-extending modules via nontrivial complex bundles and Abelian endomorphism rings," Bulletin of the Iranian Mathematical Society, vol. 43, no. 1, pp. 121-129, 2017.[7] Lam, T.Y., Lectures on modules and rings, vol. 189. Springer Science and Business Media, 2012.[8] Smith, P. F., and Tercan, A., "Direct summands of modules which satisfy (C-11)," Algebra Colloquium, vol. 11, pp. 231-237, 2004.[9] Smith, P. F., and Tercan, A., “Generalizations of CS-modules,” Communications in Algebra, vol. 21, no.6, pp.1809-1847, 1993.[10] Tercan, A. and Yücel, C. C., Module theory, extending modules and generalizations, Basel: Birkhäuser, 2016.