A new topology over the primary-like spectrum of a module

[EN] Let R be a commutative ring with identity and M a unitary R-module. The primary-like spectrum SpecL(M) is the collection of all primary-like submodules Q of  M, the recent generalization of primary ideals, such that M/Q is a primeful R-module. In this article, we topologies SpecL(M) with the patch-like topology, and show that when, SpecL(M) with the patch-like topology is a quasi-compact, Hausdorff, totally disconnected space.Rashedi, F. (2021). A new topology over the primary-like spectrum of a module. Applied General Topology. 22(2):251-257. https://doi.org/10.4995/agt.2021.13225OJS251257222M. Alkan and Y. Tiraş, Projective modules and prime submodules, Czechoslovak Math. J. 56 (2006), 601-611. https://doi.org/10.1007/s10587-006-0041-5H. Ansari-Toroghy and R. Ovlyaee-Sarmazdeh, On the prime spectrum of a module and Zariski topologies, Comm. Algebra 38 (2010), 4461-4475. https://doi.org/10.1080/00927870903386510A. Azizi, Prime submodules and flat modules, Acta Math. Sin. (Eng. Ser.) 23 (2007), 47-152. https://doi.org/10.1007/s10114-005-0813-0A. Barnard, Multiplication modules, J. Algebra 71 (1981), 174-178. https://doi.org/10.1016/0021-8693(81)90112-5J. Dauns, Prime modules, J. Reine Angew Math. 298 (1978), 156-181. https://doi.org/10.1515/crll.1978.298.156H. Fazaeli Moghimi and F. Rashedi, Zariski-like spaces of certain modules, Journal of Algebraic systems 1 (2013), 101-115.K. R. Goodearl and R. B. Warfield, An Introduction to Non-commutative Noetherian Rings (Second Edition), London Math. Soc. Student Texts 16, 2004. https://doi.org/10.1017/CBO9780511841699M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 137 (1969), 43-60. https://doi.org/10.1090/S0002-9947-1969-0251026-XC. P. Lu, A module whose prime spectrum has the surjective natural map, Houston J. Math. 33 (2007), 125-143.C. P. Lu, Saturations of submodules, Comm. Algebra 31 (2003), 2655-2673. https://doi.org/10.1081/AGB-120021886R. L. McCasland and P. F. Smith, Prime submodules of Noetherian modules, Rocky Mountain. J. Math. 23 (1993), 1041-1062. https://doi.org/10.1216/rmjm/118107254

Zariski topology on the spectrum of graded classical prime submodules

[EN] Let 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 ¿ h(R), m ¿ h(M), whenever abm ¿ N, then either am ¿ N or bm ¿ N. The spectrum of graded classical prime submodules of M is denoted by Cl.Specg(M). We topologize Cl.Specg (M) with the quasi-Zariski topology, which is analogous to that for Specg(R).Yousefian Darani, A.; Motmaen, S. (2013). Zariski topology on the spectrum of graded classical prime submodules. Applied General Topology. 14(2):159-169. doi:10.4995/agt.2013.1586.SWORD159169142S. 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.McCasland, R. L., Moore, M. E., & Smith, P. F. (1997). On the spectrum of a module over a commutative ring. Communications in Algebra, 25(1), 79-103. doi:10.1080/00927879708825840K. H. Oral, U. Tekir and A.G. Agargun, On graded prime and primary submodules, Turk. J. Math. 25, no. 3 (1999), 417-425.Roberts, P. C. (1998). Multiplicities and Chern Classes in Local Algebra. doi:10.1017/cbo9780511529986Sharp, R. Y. (1986). Asymptotic Behaviour of Certain Sets of Attached Prime Ideals. Journal of the London Mathematical Society, s2-34(2), 212-218. doi:10.1112/jlms/s2-34.2.212BAZIAR, M., & BEHBOODI, M. (2009). CLASSICAL PRIMARY SUBMODULES AND DECOMPOSITION THEORY OF MODULES. Journal of Algebra and Its Applications, 08(03), 351-362. doi:10.1142/s0219498809003369M. 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

Prime Submodules And A Sheaf On The Prime Spectra Of Modules

We define and investigate a sheaf of modules on the prime spectra of modules and it is shown that there is an isomorphism between the sections of this sheaf and the ideal transform module

A Zariski Topology for Modules

Given a duo module $M$ over an associative (not necessarily commutative) ring $R,$ a Zariski topology is defined on the spectrum $\mathrm{Spec}^{\mathrm{fp}}(M)$ of {\it fully prime} $R$-submodules of $M$. We investigate, in particular, the interplay between the properties of this space and the algebraic properties of the module under consideration.Comment: 22 pages; submitte