ABSORBING COMULTIPLICATION MODULES OVER A PULLBACK RING

Year 2018, Volume: 24 Issue: 24, 31 - 49, 05.07.2018

Atani S. Ebrahimi , Saboura Dolati Pish Hesari Mehdi Khoramdel , Maryam Sedghi Shanbeh Bazari

https://doi.org/10.24330/ieja.440164

Abstract

The purpose of this paper is to introduce the concept of 2-absorbing

comultiplication modules, which form a subclass of the class of pure-injective

modules over pullback rings. A full description of all indecomposable 2-

absorbing comultiplication modules with nite-dimensional top over the pull-

back of two discrete valuation domains with the same residue eld is given.

Keywords

Pullback, separated, non-separated, 2-absorbing comultiplication 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 pull- back rings, J. Algebra, 184(1) (1996), 304-332.
• I. Assem, D. Simson and A. Skowronski, Elements of the Representation The- ory of Associative Algebras, Vol. 1, Techniques of Representation Theory, Lon- don Math. Soc. Student Texts, 65, Cambridge University Press, Cambridge, 2006.
• [A. Badawi, On 2-absorbing ideals of commutative rings, Bull. Austral. Math. Soc., 75(3) (2007), 417-429.
• A. Badawi, U. Tekir and E. Yetkin, On 2-absorbing primary ideals in commu- tative rings, Bull. Korean Math. Soc., 51(4) (2014), 1163-1173.
• A. Badawi, U. Tekir and E. Yetkin, On weakly 2-absorbing primary ideals of commutative rings, J. Korean Math. Soc., 52(1) (2015), 97-111.
• H. Bass, On the ubiquity of Gorenstein rings, Math. Z., 82 (1963), 8-28.
• M. C. R. Butler and C. M. Ringel, Auslander-Reiten sequences with few middle terms, with applications to string algebras, Comm. Algebra, 15(1-2) (1987), 145-179.
• Ju. A. Drozd, Matrix problems and categories of matrices, in Russian, Investi- gations on the theory of representations, Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 28 (1972), 144-153.
• S. Ebrahimi Atani, On pure-injective modules over pullback rings, Comm. Al- gebra, 28(9) (2000), 4037-4069.
• S. Ebrahimi Atani, On secondary modules over Dedekind domains, Southeast Asian Bull. Math, 25(1) (2001), 1-6.
• S. Ebrahimi Atani, On secondary modules over pullback rings, Comm. Algebra, 30(6) (2002), 2675-2685.
• S. Ebrahimi Atani, Indecomposable weak multiplication modules over Dedekind domains, Demonstratio Math., 41(1) (2008), 33-43.
• R. Ebrahimi Atani and S. Ebrahimi Atani, Comultiplication modules over a pullback of Dedekind domains, Czechoslovak Math. J., 59 (2009), 1103-1114.
• R. Ebrahimi Atani and S. Ebrahimi Atani, Weak comultiplication modules over a pullback of commutative local Dedekind domains, Algebra Discrete Math., 1 (2009), 1-13.
• S. Ebrahimi Atani and F. Farzalipour, Weak multiplication modules over a pullback of Dedekind domains, Colloq. Math., 114(1) (2009), 99-112.
• S. Ebrahimi Atani and M. Sedghi Shanbeh Bazari, Absorbing multiplication modules over pullback rings, Int. Electron. J. Algebra, 21 (2017), 76-90.
• A. Facchini, Fiber products and Morita duality for commutative rings, Rend. Sem. Mat. Univ. Padova, 67 (1982), 143-159.
• A. Facchini and P. Vamos, Injective modules over pullbacks, J. London Math. Soc., 31 (1985), 425-438.
• J. Haefner and L. Klingler, Special quasi-triads and integral group rings of nite representation type I, J. Algebra, 158(2) (1993), 279-322.
• J. Haefner and L. Klingler, Special quasi-triads and integral group rings of nite representation type II, J. Algebra, 158(2) (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(1) (1990), 26-74.
• L. S. Levy, Modules over pullbacks and subdirect sums, J. Algebra, 71(1) (1981), 50-61.
• L. S. Levy, Mixed modules over ZG, G cyclic of prime order, and over related Dedekind pullbacks, J. Algebra, 71(1) (1981), 62-114.
• L. S. Levy, Modules over Dedekind-like rings, J. Algebra, 93(1) (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(1) (1997), 79-103.
• M. E. Moore and S. J. Smith, Prime and radical submodules of modules over commutative rings, Comm. Algebra, 30(10) (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 the 2-absorbing submodules, Iran. J. Math. Sci. Inform., 10(1) (2015), 131-137.
• M. Prest, Model Theory and Modules, London Mathematical Society Lecture Note Series, 130, Cambridge University Press, Cambridge, 1988.
• M. Prest, Ziegler spectra of tame hereditary algebras, J. Algebra, 207(1) (1998), 146-164.
• C. M. Ringel, Some algebraically compact modules I, Abelian groups and mod- ules (Padova, 1994), Math. Appl., 343, Kluwer Acad. Publ., Dordrecht, (1995), 419-439.
• C. M. Ringel, The Ziegler spectrum of a tame hereditary algebra, Colloq. Math., 76(1) (1998), 105-115.
• D. Simson, Linear Representations of Partially Ordered Sets and Vector Space Categories, Algebra, Logic and Applications, Vol. 4, Gordon and Breach Sci- ence Publishers, Montreux, 1992.
• D. Simson, On Corner type endo-wild algebras, J. Pure Appl. Algebra, 202(1-3) (2005), 118-132.
• R. B. Wareld, Jr., Purity and algebraic compactness for modules, Pacic J. Math., 28 (1969), 699-719.
• A. N. Wiseman, Projective modules over pullback rings, Math. Proc. Cam- bridge Philos. Soc., 97(3) (1985), 399-406.
• A. Yousean Darani and F. Soheilnia, 2-Absorbing and weakly 2-absorbing submodules, Thai J. Math., 9(3) (2011), 577-584.
• M. Ziegler, Model theory of modules, Ann. Pure Appl. Logic, 26(2) (1984), 149-213.