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Abstract
The aim of this paper is to investigate the relationship between Bézout domain, elementary divisor domain, and adequate domain. A Bézout domain is an integral domain D which every finitely generated ideal of D is principal. An integral domain D is called an elementary divisor domain if every matrix over D is equivalent to Smith normal form matrix. An adequate domain D is a Bézout domain and RP(a,b) exists for all a,b∈D with a≠0. Here the notion RP(a,b) defined as the relatively prime part of a with respect to b. It is found that every elementary divisor domain is a Bézout domain, but the converse is not true in general. It is shown the sufficient conditions for the Bézout domain being an elementary divisor domain. We also find out that every adequate domain is an elementary divisor domain. Furthermore, every one-dimensional Bézout domain is an adequate domain.
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Copyright (c) 2024 Erlangga Adinugroho Rohi, Syahida Amalia Rosyada, Sri Wahyuni

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References
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- I. Kaplansky, “Elementary divisors and modules”, Transactions of the American Mathematical Society, vol. 66, pp. 464-491, 1949. https://doi.org/10.1090/S0002-9947-1949-0031470-3
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- M.D. Walton, Bézout Domains and Elementary Divisor Domains: Are They the Same? M.Sc. [Theses]. Utah: BYU, 2023. Available: BYU ScholarsArchive.
- D.S. Dummit and R.M. Foote, Abstract Algebra, Third Edition, New Jersey: John Wiley & Sons, Inc., 2004.
- H. Chen and M.S. Abdolyousefi, “Elementary matrix reduction over Bézout domains”, Journal of Algebra and Its Applications, vol. 18, no. 18, pp. 1-12, 2019. https://doi.org/10.1142/S021949881950141X
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- M.X. Larsen, W.J. Lewis, and T.S. Shores, “Elementary divisor rings and finitely presented modules”, Transactions of the American Mathematical Society, vol. 187, pp. 231-248, 1974. https://doi.org/10.2307/1997051
- W.C. Brown, Matrices over Commutative Rings, New York: Marcel Dekker, Inc., 1993.
- C. Caldeira and J.F. Queiró, “Invariant factor of products over elementary divisor domains”, Linear Algebra and its Applications, vol. 485, pp. 345-358. https://doi.org/10.1016/j.laa.2015.07.038
- O. Helmer, “The elementary divisor theorem for certain rings without chain condition”, Bulletin of the American Mathematical Society, vol. 49, pp. 225-236, 1943. https://doi.org/10.1090/S0002-9904-1943-07886-X
- N. Mahdou and M. Zennayi, “On adequate rings”, Journal of Taibah University for Science, vol. 9, pp. 320-325, 2015. https://doi.org/10.1016/j.jtusci.2015.01.006
- L. Gillman and M. Henriksen, “Some remarks about elementary divisor rings”, Transactions of the American Mathematical Society, vol. 82, pp. 362-365, 1956. https://doi.org/10.1090/S0002-9947-1956-0078979-8
- M. Zennayi, “On adequate rings: the basic properties”, Gulf Journal of Mathematics, vol. 3, no. 3, pp. 48-56, 2015. https://doi.org/10.56947/gjom.v3i3.147
- L. Gillman and M. Henriksen, “Rings of continuous functions in which every finitely generated ideal is principal”, Transactions of the American Mathematical Society, vol. 82, no. 2, pp. 366-391, 1956. https://doi.org/10.2307/1993054
- J.W. Brewer, P.F. Conrad, and P.R. Montgomery, “Lattice-ordered groups and a conjecture for adequate domains”, Proceedings of the American Mathematical Society, vol. 43, no. 1, pp. 31-35, 1974. https://doi.org/10.2307/2039319
- R. Gilmer, Multiplicative Ideal Theory, New York: Marcel Dekker, Inc., 1972.
References
J.W. Brewer, C. Naude, and G. Naude, “On Bézout domains, elementary divisor rings, and pole assignability”, Communications in Algebra, vol. 12, no. 24, pp. 2987-3003, 1984. https://doi.org/10.1080/00927878408823140
J.W. Brewer, J.W. Bunce, and F.S. Van Vleck, Linear Systems over Commutative Rings, New York: Marcel Dekker, Inc., 1986.
J.W. Brewer, D. Katz, and W. Ullery, “On the pole assignability property over commutative rings”, Journal of Pure and Applied Algebra, vol. 48, pp. 1-7, 1985. https://doi.org/10.1016/0022-4049(87)90103-4
J.W. Brewer, D. Katz, and W. Ullery, “Pole assignability in polynomial ring, power series rings, and Prüfer domains”, Journal of Algebra, vol. 106, pp. 265-286, 1987. https://doi.org/10.1016/0021-8693(87)90031-7
E. Emre, “Regulation of linear systems over rings by dynamic output feedback”, Systems & Control Letters, vol. 3, pp. 57-62, 1983. https://doi.org/10.1016/0167-6911(83)90039-7
E. Emre and P.P. Khargonekar, “Pole placement for linear systems over Bézout domains”, IEEE Transactions on Automatic Control, vol. AC-29, no. 1, pp. 90-91, 1984. https://doi.org/10.1109/TAC.1984.1103371
I. Kaplansky, “Elementary divisors and modules”, Transactions of the American Mathematical Society, vol. 66, pp. 464-491, 1949. https://doi.org/10.1090/S0002-9947-1949-0031470-3
B. Zabavsky and A. Gatalevych, “A commutative Bézout domain PM* domain is an elementary divisor ring”, Algebra and Discrete Mathematics, vol. 19, no. 2, pp. 295-301, 2015.
M.D. Walton, Bézout Domains and Elementary Divisor Domains: Are They the Same? M.Sc. [Theses]. Utah: BYU, 2023. Available: BYU ScholarsArchive.
D.S. Dummit and R.M. Foote, Abstract Algebra, Third Edition, New Jersey: John Wiley & Sons, Inc., 2004.
H. Chen and M.S. Abdolyousefi, “Elementary matrix reduction over Bézout domains”, Journal of Algebra and Its Applications, vol. 18, no. 18, pp. 1-12, 2019. https://doi.org/10.1142/S021949881950141X
D. Lorenzini, “Elementary divisor domains and Bézout domains”, Journal of Algebra, vol. 371, pp. 609-619, 2012. https://doi.org/10.1016/j.jalgebra.2012.08.020
B.V. Zabavsky, O.V. Domsha, and O.M. Romaniv, “Clear rings and clear elements”, Matematychni Studii, vol. 55, no. 1, pp. 3-9, 2021. https://doi.org/10.30970/ms.55.1.3-9
M.X. Larsen, W.J. Lewis, and T.S. Shores, “Elementary divisor rings and finitely presented modules”, Transactions of the American Mathematical Society, vol. 187, pp. 231-248, 1974. https://doi.org/10.2307/1997051
W.C. Brown, Matrices over Commutative Rings, New York: Marcel Dekker, Inc., 1993.
C. Caldeira and J.F. Queiró, “Invariant factor of products over elementary divisor domains”, Linear Algebra and its Applications, vol. 485, pp. 345-358. https://doi.org/10.1016/j.laa.2015.07.038
O. Helmer, “The elementary divisor theorem for certain rings without chain condition”, Bulletin of the American Mathematical Society, vol. 49, pp. 225-236, 1943. https://doi.org/10.1090/S0002-9904-1943-07886-X
N. Mahdou and M. Zennayi, “On adequate rings”, Journal of Taibah University for Science, vol. 9, pp. 320-325, 2015. https://doi.org/10.1016/j.jtusci.2015.01.006
L. Gillman and M. Henriksen, “Some remarks about elementary divisor rings”, Transactions of the American Mathematical Society, vol. 82, pp. 362-365, 1956. https://doi.org/10.1090/S0002-9947-1956-0078979-8
M. Zennayi, “On adequate rings: the basic properties”, Gulf Journal of Mathematics, vol. 3, no. 3, pp. 48-56, 2015. https://doi.org/10.56947/gjom.v3i3.147
L. Gillman and M. Henriksen, “Rings of continuous functions in which every finitely generated ideal is principal”, Transactions of the American Mathematical Society, vol. 82, no. 2, pp. 366-391, 1956. https://doi.org/10.2307/1993054
J.W. Brewer, P.F. Conrad, and P.R. Montgomery, “Lattice-ordered groups and a conjecture for adequate domains”, Proceedings of the American Mathematical Society, vol. 43, no. 1, pp. 31-35, 1974. https://doi.org/10.2307/2039319
R. Gilmer, Multiplicative Ideal Theory, New York: Marcel Dekker, Inc., 1972.