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Characterizations of Gelfand rings specially clean rings and their dual rings. (English) Zbl 1448.14001

Authors’ abstract: In this paper, new criteria for zero dimensional rings, Gelfand rings, clean rings and mp-rings are given. A new class of rings is introduced and studied, we call them purified rings. Specially, reduced purified rings are characterized. New characterizations for pure ideals of reduced Gelfand rings and mp-rings are provided. It is also proved that if the topology of a scheme is Hausdorff, then the affine opens of that scheme is stable under taking finite unions. In particular, every compact scheme is an affine scheme.

MSC:

14A05 Relevant commutative algebra
14A15 Schemes and morphisms
14R05 Classification of affine varieties
14R10 Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem)
13A99 General commutative ring theory
13B10 Morphisms of commutative rings
13E05 Commutative Noetherian rings and modules
13H99 Local rings and semilocal rings
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References:

[1] Al-Ezeh, H., Pure ideals in commutative reduced Gelfand rings with unity, Arch. Math., 53, 266-269 (1989) · Zbl 0667.13001 · doi:10.1007/BF01277063
[2] Al-Ezeh, H., The pure spectrum of a PF-ring, Comment. Math. Univ. St. Paul, 37, 2, 179-183 (1988) · Zbl 0683.13001
[3] Al-Ezeh, H., Exchange pf-rings and almost pp-rings, Int. J. Math. Math. Sci., 12, 4, 725-728 (1989) · Zbl 0692.13007 · doi:10.1155/S016117128900089X
[4] Anderson, DD; Camillo, VP, Commutative rings whose elements are a sum of a unit and idempotent, Commun. Algebra, 30, 7, 3327-3336 (2002) · Zbl 1083.13501 · doi:10.1081/AGB-120004490
[5] Ara, P., Gromov translation algebras over dicrete trees are exchange rings, Trans. Am. Math. Soc., 356, 5, 2067-2079 (2004) · Zbl 1058.16008 · doi:10.1090/S0002-9947-03-03372-5
[6] Artico, G.; Marconi, U., On the compactness of minimal spectrum, Rend. Semin. Mat. Univ. Padova, 56, 79-84 (1976) · Zbl 0381.13003
[7] Bhattacharjee, P.; McGovern, WW, When \({{\rm Min}}(A)^{-1}\) is Hausdorff, Commun. Algebra, 41, 99-108 (2013) · Zbl 1264.13004 · doi:10.1080/00927872.2011.617228
[8] Borceux, F.; Van den Bossche, G., Algebra in a Localic Topos with Applications to Ring Theory. Lecture Notes in Mathematics (1983), Berlin: Springer, Berlin · Zbl 0522.18001
[9] Chapman, S., Multiplicative Ideal Theory and Factorization Theory: Commutative and Non-commutative Perspectives (2016), Berlin: Springer, Berlin
[10] Contessa, M., On pm-rings, Commun. Algebra, 10, 93-108 (1982) · Zbl 0484.13002 · doi:10.1080/00927878208822703
[11] Contessa, M., On cerrain classes of pm-rings, Commun. Algebra, 12, 1447-1469 (1984) · Zbl 0545.13001 · doi:10.1080/00927878408823063
[12] Contessa, M., Ultraproducts of pm-rings and mp-rings, J. Pure Appl. Algebra, 32, 11-20 (1984) · Zbl 0539.13003 · doi:10.1016/0022-4049(84)90010-0
[13] Couchot, F., Indecomposable modules and Gelfand rings, Commun. Algebra, 35, 231-241 (2007) · Zbl 1107.13012 · doi:10.1080/00927870601041615
[14] de Jong A.J., et al.: Stacks Project, see http://stacks.math.columbia.edu
[15] De Marco, G.; Orsatti, A., Commutative rings in which every prime ideal is contained in a unique maximal ideal, Proc. Am. Math. Soc., 30, 3, 459-466 (1971) · Zbl 0207.05001 · doi:10.1090/S0002-9939-1971-0282962-0
[16] Engelking, R.: General Topology, Revised and completed edition. Heldermann, Berlin (1989) · Zbl 0684.54001
[17] Gilmer, R., Background and Preliminaries on Zero-Dimensional Rings, Zero-Dimensional Commutative Rings, 1-13 (1995), New York: Marcel Dekker, New York · Zbl 0882.13011
[18] Goodearl, KR; Warfield, RB, Algebras over zero-dimensional rings, Math. Ann., 223, 157-168 (1976) · Zbl 0317.16004 · doi:10.1007/BF01360879
[19] Hartshorne, R., Algebraic Geometry (1977), Berlin: Springer, Berlin · Zbl 0367.14001
[20] Huckaba, JA, Commutative Rings with Zero Divisors (1988), New York: Marcel Dekker Inc., New York · Zbl 0637.13001
[21] Liu, Q., Algebraic Geometry and Arithmetic Curves (2002), New York: Oxford University Press Inc., New York · Zbl 0996.14005
[22] McGovern, WW, Neat rings, J. Pure Appl. Algebra, 205, 243-265 (2006) · Zbl 1095.13025 · doi:10.1016/j.jpaa.2005.07.012
[23] Nicholson, WK, Lifting idempotents and exchange rings, Trans. Am. Math. Soc., 229, 269-278 (1977) · Zbl 0352.16006 · doi:10.1090/S0002-9947-1977-0439876-2
[24] Simmons, H., Reticulated rings, J. Algebra, 66, 1, 169-192 (1980) · Zbl 0462.13002 · doi:10.1016/0021-8693(80)90118-0
[25] Simmons, H., Erratum, J. Algebra, 74, 1, 292 (1982) · Zbl 0475.13002 · doi:10.1016/0021-8693(82)90024-2
[26] Tarizadeh, A., Flat topology and its dual aspects, Commun. Algebra, 47, 1, 195-205 (2019) · Zbl 1410.13001 · doi:10.1080/00927872.2018.1469637
[27] Tarizadeh, A., Notes on finitely generated flat modules, Bull. Korean Math. Soc., 57, 2, 419-427 (2020) · Zbl 1440.13047
[28] Tarizadeh, A., Zariski compactness of minimal spectrum and flat compactness of maximal spectrum, J. Algebra Appl., 18, 11, 1950202 (2019) · Zbl 1425.13001 · doi:10.1142/S0219498819502025
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