Zariski topology on the spectrum of graded classical prime submodules
Keywords:Graded prime ideal, Zariski topology, Quasi-Zariski topology
AbstractLet $R$ be a $G$-graded commutative ring with identity and let $M$ be a graded $R$-module. A proper graded submodule $N$ of $M$ is called graded classical prime if for every $a, b\in h(R)$, $m\in h(M)$, whenever $abm\in N$, then either $am\in N$ or $bm\in N$. The spectrum of graded classical prime submodules of $M$ is denoted by $Cl.Spec_g(M)$. We topologize $Cl.Spec_g(M)$ with the quasi-Zariski topology, which is analogous to that for $Spec_g(R)$.
S. Ebrahimi Atani and F. Farzalipour, On weakly prime submodules, Tamkang Journal of Mathematics 38, no. 3 (2007), 247-252.
S. Ebrahimi Atani and F. Farzalipour, On graded multiplication modules, Chiang-Mai Journal of Science, to appear.
S. Ebrahimi Atani and F.E.K. Saraei, Graded modules which satisfy the Gr-Radical formola, Thai Journal of Mathematics 8, no. 1 (2010), 161-170.
P. Lu, The Zariski topology on the prime spectrum of a module, Houston J. Math. 25, no. 3 (1999), 417-425.
R. L. McCasland, M. E. Moore and P. F. Smith, On the spectrum of a module over a commutative ring, Comm. Algebra 25, no. 1 (1997), 79-103. http://dx.doi.org/10.1080/00927879708825840
K. H. Oral, U. Tekir and A.G. Agargun, On graded prime and primary submodules, Turk. J. Math. 25, no. 3 (1999), 417-425.
P. C. Roberts, Multiplicities and Chern classes in local algebra, Cambridge University Press, 1998. http://dx.doi.org/10.1017/CBO9780511529986
R. Y. Sharp, Asymptotic behaviour of certain sets of attached prime ideals, J. London Math. Soc. 34, no. 2 (1986), 212-218. http://dx.doi.org/10.1112/jlms/s2-34.2.212
Yousefia Darani, Topologies on $Spec_g(M)$, Buletinul Academiei de Stiinte a Republicii Moldova Matematica, to appear.
M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Longman Higher Education, New York 1969.
M. Baziar and M. Behboodi, Classical primary submodules and decomposition theory of modules, J. Algebra Appl. 8, no. 3 (2009), 351–362. http://dx.doi.org/10.1142/S0219498809003369
M. Behboodi and H. Koohi, Weakly prime modules, Vietnam J. Math. 32, no. 2 (2004), 185–195.
M. Behboodi and M. J. Noori, Zariski-Like topology on the classical prime spectrum of a module, Bull. Iranian Math. Soc. 35, no. 1 (2009), 255–271.
M. Behboodi and S. H. Shojaee, On chains of classical prime submodules and dimension theory of modules, Bulletin of the Iranian Mathematical Society 36 (2010), 149–166.
J. Dauns, Prime modules, J. Reine Angew. Math. 298 (1978), 156–181.
S. Ebrahimi Atani, On graded prime submodules, Chiang Mai J. Sci. 33, no. 1 (2006), 3–7.
How to Cite
This journal is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.