ON QUASI COMULTIPLICATION MODULES OVER PULLBACK RINGS
Year 2019,
Volume: 26 Issue: 26, 95 - 110, 11.07.2019
S. Ebrahimi Atani
,
F. Esmaeili Khalil Saraei
Abstract
We classify all indecomposable quasi comultiplication modules
over pullback of two Dedekind domains. We extend the denitions and the
results of comultiplication modules over pullback rings to a more general quasi
comultiplication modules case.
References
- A. Abbasi and D. Hassanzadeh-Lelekaami, Quasi-prime submodules and developed Zariski topology, Algebra Colloq., 19 (2012), 1089-1108.
- 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, Techniques of Representation Theory, London
Mathematical Society Student Texts, 65, Cambridge University Press,
Cambridge, 2006.
- H. Bass, On the ubiquity of Gorenstein rings, Math. Z., 82 (1963), 8-28.
- S. Ebrahimi Atani, On pure-injective modules over pullback rings, Comm. Algebra,
28 (2000), 4037-4069.
- S. Ebrahimi Atani, On secondary modules over pullback rings, Comm. Algebra,
30 (2002), 2675-2685.
- 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.
- R. Ebrahimi Atani and S. Ebrahimi Atani, On primarily multiplication modules
over pullback rings, Algebra Discrete Math., 11 (2011), 1-17.
- S. Ebrahimi Atani and F. Esmaeili Khalil Saraei, Indecomposable primarily
comultiplication modules over a pullback of two Dedekind domains, Colloq.
Math., 120 (2010), 23-42.
- S. Ebrahimi Atani and F. Esmaeili Khalil Saraei, On P-torsion modules over
a pullback of two Dedekind domains, Georgian Math. J., 19 (2012), 473-488.
- S. Ebrahimi Atani, S. Dolati Pish Hesari, M. Khoramdel and M. Sedghi Shanbeh
Bazari, Absorbing comultiplication modules over a pullback ring, Int. Electron.
J. Algebra, 24 (2018), 31-49.
- F. Esmaeili Khalil Saraei, On pseudo-prime multiplication modules over pullback rings, Colloq. Math., 143 (2016), 63-78.
- 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.
- W. J. Heinzer, L. J. Ratlif, Jr. and D. E. Rush, Strongly irreducible ideals of a
commutative ring, J. Pure Appl. Algebra, 166 (2002), 267-275.
- R. Kielpinski, On Gamma-pure injective modules, Bull. Acad. Polon. Sci. Ser. Sci.
Math. Astronom. Phys., 15 (1967), 127-131.
- V. V. Kiricenko, Classication of the pairs of mutually annihilating operators in
a graded space and representations of a dyad of generalized uniserial algebras,
In: \Rings and Linear Groups", Zap. Naucn. Sem. Leningrad. Otdel. Mat.
Inst. Steklov. (LOMI), 75 (1978), 91-109, 196-197.
- L. Klingler, Integral representations of groups of square-free order, J. Algebra,
129 (1990), 26-74.
- 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 pullbacks and subdirect sums, J. Algebra, 71 (1981),
50-61.
- L. S. Levy, Modules over Dedekind-like rings, J. Algebra, 93 (1985), 1-116.
- 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. Acad. Nauk SSSR Ser. Mat., 33 (1969), 65-89.
- M. Prest, Model Theory and Modules, London Mathematical Society Lecture
Note Series, 130, Cambridge University Press, Cambridge, 1988.
- D. Simson, Linear Representations of Partially Ordered Sets and Vector Space
Categories, Algebra, Logic and Applications, 4, Gordon and Breach Science
Publishers, Switzerland-Australia, 1992.
- D. Simson and A. Skowronski, Elements of the Representation Theory of Associative
Algebras, Representation-Innite Tilted Algebras, Vol. 3, London
Mathematical Society Student Texts, 72, Cambridge University Press, Cambridge,
2007.
- R. B. Wareld, Jr., Purity and algebraic compactness for modules, Pacic J.
Math., 28 (1969), 699-719.
- A. N. Wiseman, Projective modules over pullback rings, Math. Proc. Cambridge
Philos. Soc., 97 (1985), 399-406.
Year 2019,
Volume: 26 Issue: 26, 95 - 110, 11.07.2019
S. Ebrahimi Atani
,
F. Esmaeili Khalil Saraei
References
- A. Abbasi and D. Hassanzadeh-Lelekaami, Quasi-prime submodules and developed Zariski topology, Algebra Colloq., 19 (2012), 1089-1108.
- 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, Techniques of Representation Theory, London
Mathematical Society Student Texts, 65, Cambridge University Press,
Cambridge, 2006.
- H. Bass, On the ubiquity of Gorenstein rings, Math. Z., 82 (1963), 8-28.
- S. Ebrahimi Atani, On pure-injective modules over pullback rings, Comm. Algebra,
28 (2000), 4037-4069.
- S. Ebrahimi Atani, On secondary modules over pullback rings, Comm. Algebra,
30 (2002), 2675-2685.
- 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.
- R. Ebrahimi Atani and S. Ebrahimi Atani, On primarily multiplication modules
over pullback rings, Algebra Discrete Math., 11 (2011), 1-17.
- S. Ebrahimi Atani and F. Esmaeili Khalil Saraei, Indecomposable primarily
comultiplication modules over a pullback of two Dedekind domains, Colloq.
Math., 120 (2010), 23-42.
- S. Ebrahimi Atani and F. Esmaeili Khalil Saraei, On P-torsion modules over
a pullback of two Dedekind domains, Georgian Math. J., 19 (2012), 473-488.
- S. Ebrahimi Atani, S. Dolati Pish Hesari, M. Khoramdel and M. Sedghi Shanbeh
Bazari, Absorbing comultiplication modules over a pullback ring, Int. Electron.
J. Algebra, 24 (2018), 31-49.
- F. Esmaeili Khalil Saraei, On pseudo-prime multiplication modules over pullback rings, Colloq. Math., 143 (2016), 63-78.
- 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.
- W. J. Heinzer, L. J. Ratlif, Jr. and D. E. Rush, Strongly irreducible ideals of a
commutative ring, J. Pure Appl. Algebra, 166 (2002), 267-275.
- R. Kielpinski, On Gamma-pure injective modules, Bull. Acad. Polon. Sci. Ser. Sci.
Math. Astronom. Phys., 15 (1967), 127-131.
- V. V. Kiricenko, Classication of the pairs of mutually annihilating operators in
a graded space and representations of a dyad of generalized uniserial algebras,
In: \Rings and Linear Groups", Zap. Naucn. Sem. Leningrad. Otdel. Mat.
Inst. Steklov. (LOMI), 75 (1978), 91-109, 196-197.
- L. Klingler, Integral representations of groups of square-free order, J. Algebra,
129 (1990), 26-74.
- 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 pullbacks and subdirect sums, J. Algebra, 71 (1981),
50-61.
- L. S. Levy, Modules over Dedekind-like rings, J. Algebra, 93 (1985), 1-116.
- 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. Acad. Nauk SSSR Ser. Mat., 33 (1969), 65-89.
- M. Prest, Model Theory and Modules, London Mathematical Society Lecture
Note Series, 130, Cambridge University Press, Cambridge, 1988.
- D. Simson, Linear Representations of Partially Ordered Sets and Vector Space
Categories, Algebra, Logic and Applications, 4, Gordon and Breach Science
Publishers, Switzerland-Australia, 1992.
- D. Simson and A. Skowronski, Elements of the Representation Theory of Associative
Algebras, Representation-Innite Tilted Algebras, Vol. 3, London
Mathematical Society Student Texts, 72, Cambridge University Press, Cambridge,
2007.
- R. B. Wareld, Jr., Purity and algebraic compactness for modules, Pacic J.
Math., 28 (1969), 699-719.
- A. N. Wiseman, Projective modules over pullback rings, Math. Proc. Cambridge
Philos. Soc., 97 (1985), 399-406.