Pseudo-absorbing comultiplication modules over a pullback ring

Year 2024, Volume: 35 Issue: 35, 1 - 19, 09.01.2024

Shahabaddin Ebrahimi Atanı Mehdi Khoramdel Saboura Dolatı Pıshhesarı

https://doi.org/10.24330/ieja.1404416

Abstract

In this paper, we introduce the notion of pseudo-absorbing
comultiplication modules. A full description of all indecomposable pseudo-absorbing
comultiplication modules with finite dimensional top over certain kinds of pullback rings
are given and establish a connection between the pseudo-absorbing comultiplication
modules and the pure-injective modules over such rings.

Keywords

Pullback, separated, non-separated, pseudo-absorbing comultiplication module, pure-injective module

References

• Y. Al-Shania  and P. F. Smith, Comultiplication modules over commutative rings, J. Commut. Algebra, 3(1) (2011), 1-29.
• D. M. Arnold and R. C. Laubenbacher, Finitely generated modules over pullback rings, J. Algebra, 184 (1996), 304-332.
• I. Assem, D. Simson and A. Skowronski, Elements of the Representation Theory of Associative Algebras, Vol. 1, London Math. Soc. Stud. Texts, 65, Cambridge University Press, Cambridge, 2006.
• A. Badawi, On 2-absorbing ideals of commutative rings, Bull. Austral. Math. Soc., 75 (2007), 417-429.
• A. Badawi, U. Tekir and E. Yetkin, On 2-absorbing primary ideals in commutative rings, Bull. Korean Math. Soc., 51 (2014), 1163-1173.
• A. Badawi, U. Tekir and E. Yetkin, On weakly 2-absorbing primary ideals of commutative rings, J. Korean Math. Soc., 52 (2015), 97-111.
• S. Ebrahimi Atani, On pure-injective modules over pullback rings, Comm. Algebra, 28 (2000), 4037-4069.
• S. Ebrahimi Atani, On secondary modules over Dedekind domains, Southeast Asian Bull. Math, 25 (2001), 1-6.
• S. Ebrahimi Atani, On secondary modules over pullback rings, Comm. Algebra, 30 (2002), 2675-2685.
• S. Ebrahimi Atani, Indecomposable weak multiplication modules over Dedekind domains, Demonstratio Math., 41 (2008), 33-43.
• S. Ebrahimi Atani and F. Farzalipour, Weak multiplication modules over a pullback of Dedekind domains, Colloq. Math., 114 (2009), 99-112.
• S. Ebrahimi Atani, S. Dolati Pish Hesari, M. Khoramdel and M. Sedghi Shanbeh Bazari, Pseudo-absorbing multiplication modules over a pullback ring, J. Pure Appl. Algebra, 222 (2018), 3124-3136.
• A. Facchini and P. Vamos, Injective modules over pullbacks, J. London Math. Soc. (2), 31 (1985), 425-438.
• J. Haefner and L. Klingler, Special quasi-triads and integral group rings of finite representation type I, J. Algebra, 158 (1993), 279-322.
• J. Haefner and L. Klingler, Special quasi-triads and integral group rings of finite representation type II, J. Algebra, 158 (1993), 323-374.
• I. Kaplansky, Modules over Dedekind rings and valuation rings, Trans. Amer. Math. Soc., 72 (1952), 327-340.
• L. Klingler, Integral representations of groups of square-free order, J. Algebra, 129 (1990), 26-74.
• P. A. Krylov and A. A. Tuganbaev, Modules over Discrete Valuation Rings, De Gruyter Expositions in Mathematics, Berlin, 2018.
• L. S. Levy, Modules over pullbacks and subdirect sums, J. Algebra, 71 (1981), 50-61.
• L. S. Levy, Mixed modules over ZG, G cyclic of prime order, and over related Dedekind pullbacks, J. Algebra, 71 (1981), 62-114.
• L. S. Levy, Modules over Dedekind-like rings, J. Algebra, 93 (1985), 1-116.
• R. L. McCasland, M. E. Moore and P. F. Smith, On the spectrum of a module over a commutative ring, Comm. Algebra, 25 (1997), 79-103.
• M. E. Moore and S. J. Smith, Prime and radical submodules of modules over commutative rings, Comm. Algebra, 30 (2002), 5037-5064.
• L. A. Nazarova and A. V. Roiter, Finitely generated modules over a dyad of two local Dedekind rings, and  nite groups which possess an abelian normal divisor of index p, Izv. Akad. Nauk SSSR Ser. Mat., 33 (1969), 65-89.
• Sh. Payrovi and S. Babaei, On 2-absorbing submodules, Algebra Colloq., 19(1) (2012), 913-920.
• M. Prest, Model Theory and Modules, London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1988.
• M. Prest, Ziegler spectra of tame hereditary algebras, J. Algebra, 207 (1998), 146-164.
• C. M. Ringel, Some algebraically compact modules I, Abelian Groups and Modules, Math. Appl., vol. 343, Padova, 1994, Kluwer Acad. Publ., Dordrecht (1995), 419-439.
• C. M. Ringel, The Ziegler spectrum of a tame hereditary algebra, Colloq. Math., 76 (1998), 105-115.
• D. Simson, Linear Representations of Partially Ordered Sets and Vector Space Categories, Algebra Logic Appl., Gordon and Breach Science Publishers, Montreux, 1992.
• Y. Wang and Y. Liu, A note on comultiplication modules, Algebra Colloq., 21(1) (2014), 147-150.
• R. B.Warfield, Purity and algebraic compactness for modules, Pacific J. Math., 28 (1969), 699-719.