Research Article
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Year 2018, , 31 - 49, 05.07.2018
https://doi.org/10.24330/ieja.440164

Abstract

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.

ABSORBING COMULTIPLICATION MODULES OVER A PULLBACK RING

Year 2018, , 31 - 49, 05.07.2018
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.

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.
There are 40 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Atani S. Ebrahimi

Saboura Dolati Pish Hesari This is me

Mehdi Khoramdel

Maryam Sedghi Shanbeh Bazari

Publication Date July 5, 2018
Published in Issue Year 2018

Cite

APA S. Ebrahimi, A., Hesari, S. D. P., Khoramdel, M., Bazari, M. S. S. (2018). ABSORBING COMULTIPLICATION MODULES OVER A PULLBACK RING. International Electronic Journal of Algebra, 24(24), 31-49. https://doi.org/10.24330/ieja.440164
AMA S. Ebrahimi A, Hesari SDP, Khoramdel M, Bazari MSS. ABSORBING COMULTIPLICATION MODULES OVER A PULLBACK RING. IEJA. July 2018;24(24):31-49. doi:10.24330/ieja.440164
Chicago S. Ebrahimi, Atani, Saboura Dolati Pish Hesari, Mehdi Khoramdel, and Maryam Sedghi Shanbeh Bazari. “ABSORBING COMULTIPLICATION MODULES OVER A PULLBACK RING”. International Electronic Journal of Algebra 24, no. 24 (July 2018): 31-49. https://doi.org/10.24330/ieja.440164.
EndNote S. Ebrahimi A, Hesari SDP, Khoramdel M, Bazari MSS (July 1, 2018) ABSORBING COMULTIPLICATION MODULES OVER A PULLBACK RING. International Electronic Journal of Algebra 24 24 31–49.
IEEE A. S. Ebrahimi, S. D. P. Hesari, M. Khoramdel, and M. S. S. Bazari, “ABSORBING COMULTIPLICATION MODULES OVER A PULLBACK RING”, IEJA, vol. 24, no. 24, pp. 31–49, 2018, doi: 10.24330/ieja.440164.
ISNAD S. Ebrahimi, Atani et al. “ABSORBING COMULTIPLICATION MODULES OVER A PULLBACK RING”. International Electronic Journal of Algebra 24/24 (July 2018), 31-49. https://doi.org/10.24330/ieja.440164.
JAMA S. Ebrahimi A, Hesari SDP, Khoramdel M, Bazari MSS. ABSORBING COMULTIPLICATION MODULES OVER A PULLBACK RING. IEJA. 2018;24:31–49.
MLA S. Ebrahimi, Atani et al. “ABSORBING COMULTIPLICATION MODULES OVER A PULLBACK RING”. International Electronic Journal of Algebra, vol. 24, no. 24, 2018, pp. 31-49, doi:10.24330/ieja.440164.
Vancouver S. Ebrahimi A, Hesari SDP, Khoramdel M, Bazari MSS. ABSORBING COMULTIPLICATION MODULES OVER A PULLBACK RING. IEJA. 2018;24(24):31-49.