Geodesic
Arithmetical rings
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Algebra, Tome 164 (2019), pp. 3-73.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper, some familiar and new results on arithmetical rings, modules, and Besout rings (not necessarily commutative) are provided. In particular, we examine relationships between arithmetical rings and their localizations by maximal ideals, saturated submodules and saturations, localizable rings, properties of annihilators of finitely generated modules over arithmetical rings, diagonalizable rings, rings with flat right ideals, and rings with quasi-projective finitely generated right ideals, Hermite rings, Pierce stalks, and rings with Krull dimension.
Keywords: arithmetic ring, flat module, localization by maximal ideal, Bezout ring, Hermite ring, diagonalizable ring, Pierce stalk.
Mots-clés : distribution module 
@article{INTO_2019_164_a0,
     author = {A. A. Tuganbaev},
     title = {Arithmetical rings},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {3--73},
     publisher = {mathdoc},
     volume = {164},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2019_164_a0/}
}
TY  - JOUR
AU  - A. A. Tuganbaev
TI  - Arithmetical rings
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2019
SP  - 3
EP  - 73
VL  - 164
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/INTO_2019_164_a0/
LA  - ru
ID  - INTO_2019_164_a0
ER  - 
%0 Journal Article
%A A. A. Tuganbaev
%T Arithmetical rings
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2019
%P 3-73
%V 164
%I mathdoc
%U http://geodesic.mathdoc.fr/item/INTO_2019_164_a0/
%G ru
%F INTO_2019_164_a0
A. A. Tuganbaev. Arithmetical rings. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Algebra, Tome 164 (2019), pp. 3-73. http://geodesic.mathdoc.fr/item/INTO_2019_164_a0/

[1] Vechtomov E. M., “Distributivnye koltsa nepreryvnykh funktsii i $F$-prostranstva”, Mat. zametki., 34:3 (1983), 321–332 | MR | Zbl

[2] Golod E. S., “Arifmeticheskie koltsa, biendomorfizmy i bazisy Grebnera”, Usp. mat. nauk., 60:1 (2005), 167–168 | DOI | MR | Zbl

[3] Golod E. S., “Distributivnost, binarnye sootnosheniya i standartnye bazisy”, Fundam. prikl. mat., 16:3 (2010), 127–134

[4] Golod E. S., “Zamechanie o kommutativnykh arifmeticheskikh koltsakh”, Fundam. prikl. mat., 19:2 (2014), 21–23

[5] Golod E. S., Tuganbaev A. A., “Annulyatory i konechno porozhdennye moduli”, Fundam. prikl. mat., 21:1 (2016), 79–82 | MR

[6] Kirichenko V. V., Yaremenko Yu. V., “O polusovershennykh poludistributivnykh koltsakh”, Mat. zametki., 69:1 (2001), 153–156 | DOI | MR | Zbl

[7] Tuganbaev A. A., “Kharakterizatsii kolets, ispolzuyuschie maloin'ektivnye i maloproektivnye moduli”, Vestn. MGU. Ser. mat. mekh., 1979, no. 3, 48–51 | Zbl

[8] Tuganbaev A. A., “Distributivnye moduli”, Usp. mat. nauk., 35:5 (215) (1980), 245–246 | MR | Zbl

[9] Tuganbaev A. A., “Koltsa s ploskimi pravymi idealami i distributivnye koltsa”, Mat. zametki., 38:2 (1985), 218–228 | MR | Zbl

[10] Tuganbaev A. A., “Ploskie moduli i koltsa, konechno porozhdennye kak moduli nad svoim tsentrom”, Mat. zametki., 60:2 (1996), 254–277 | DOI | MR | Zbl

[11] Tuganbaev A. A., “Distributivnye nelokalizuemye koltsa”, Usp. mat. nauk., 57:3 (2002), 165–166 | DOI | MR | Zbl

[12] Tuganbaev A. A., “Stroenie distributivnykh kolets”, Mat. sb., 193:5 (2002), 113–128 | DOI | MR | Zbl

[13] Tuganbaev A. A., “Distributivnye i multiplikatsionnye moduli i koltsa”, Mat. zametki., 75:3 (2004), 425–434 | DOI | MR | Zbl

[14] Tuganbaev A. A., “Arifmeticheskie koltsa i kvaziproektivnye idealy”, Fundam. prikl. mat., 19:2 (2014), 207–211

[15] Tuganbaev A. A., “Koltsa, u kotorykh vse konechnoporozhdennye pravye idealy kvaziproektivny”, Diskr. mat., 27:1 (2015), 146–154 | DOI | Zbl

[16] Tuganbaev A. A., “Koltsa Bezu bez netsentralnykh idempotentov”, Diskr. mat., 28:2 (2016), 133–145 | DOI | MR | Zbl

[17] Tuganbaev A. A., “Koltsa Bezu, annulyatory i diagonaliziruemost”, Fundam. prikl. mat., 21:2 (2016), 253–256 | MR

[18] Tuganbaev A. A., “Arifmeticheskie koltsa i razmernost Krullya”, Diskr. mat., 29:3 (2017), 126–132 | DOI

[19] Abuhlail J., Jarrar M., Kabbaj S., “Commutative rings in which every finitely generated ideal is quasi-projective”, J. Pure Appl. Algebra., 215 (2011), 2504–2511 | DOI | MR | Zbl

[20] Achkar H., “Sur les anneaux arithmétiques à gauche”, C. R. Acad. Sci., 278:5 (1974), A307–A309 | MR

[21] Achkar H., “Sur les propriétés des anneaux arithmétiques à gauche”, C. R. Acad. Sci., 284:17 (1977), A993–A995 | MR

[22] Achkar H., “Anneaux arithmétiques à gauche”, C. R. Acad. Sci., 286:20 (1978), A871–A873 | MR

[23] Achkar H., “Anneaux arithmétiques à gauche: propriétés de transfert”, C. R. Acad. Sci., 288:11 (1979), A583–A586 | MR

[24] Albu T., Năstăsescu C., “Modules arithmétiques”, Acta. Math. Acad. Sci. Hung., 25:3–4 (1974), 299–311 | DOI | MR | Zbl

[25] Amitsur S. A., “Remarks on principal ideal rings”, Osaka Math. J., 15:1 (1963), 59–69 | MR | Zbl

[26] Anderson D. D., “Some remarks on the ring $R(X)$”, Comment. Math. Univ. St. Pauli., 26:2 (1977), 137–140 | MR

[27] Anderson D. D., Anderson D. F., Markanda R., “The rings $R(X)$ and $R\langle X\rangle $”, J. Algebra., 95:1 (1985), 96–115 | DOI | MR | Zbl

[28] Anderson D. D., Camillo V., “Armendariz rings and Gaussian rings”, Commun. Algebra., 26:7 (1998), 2265–2272 | DOI | MR | Zbl

[29] Anderson F., Fuller K. R., Rings and Categories of Modules, Springer-Verlag, Berlin, 1973 | MR

[30] Aubert K. E., Beck I., “Chinese rings”, J. Pure Appl. Algebra., 24 (1982), 221–226 | DOI | MR | Zbl

[31] Bandelt H. J., Petrich M., “Subdirect products of rings and distributive lattices”, Proc. Edinburgh Math. Soc., 25:2 (1982), 155–171 | DOI | MR | Zbl

[32] Bazzoni S., Glaz S., “Prüfer rings”, Multiplicative Ideal Theory in Commutative Algebra, eds. Brewer J. W., Glaz S., Heinzer W. J., Olberding B. M., Springer, Boston, 2006 | MR | Zbl

[33] Bazzoni S., Glaz S., “Gaussian properties of total rings of quotients”, J. Algebra., 310 (2007), 180–193 | DOI | MR | Zbl

[34] Behrens E. A., “Distributive Darstellbare Ringe, I”, Math. Z., 73:5 (1960), 409–432 | DOI | MR | Zbl

[35] Behrens E. A., “Distributive Darstellbare Ringe, II”, Math. Z., 76:4 (1961), 367–384 | DOI | MR | Zbl

[36] Behrens E. A., “Die Halbgruppe der Ideale in Algebren mit distributivem Idealverband”, Arch. Math., 13 (1962), 251–266 | DOI | MR | Zbl

[37] Behrens E. A., Ring Theory, Acad. Press, New York, 1972 | MR

[38] Behrens E. A., “Noncommutative arithmetic rings”, Rings, Modules and Radicals, London, Amsterdam, 1973, 67–71 | MR

[39] Belzner T., “Towards self-duality of semidistributive Artinian rings”, J. Algebra., 135:1 (1990), 74–95 | DOI | MR | Zbl

[40] Bessenrodt K., Brungs H. H., Törner G., “Right chain rings, Part 1”, Schriftenreihe des Fachbereich Mathematik, Universität Duisburg, 1990

[41] Blair R. L., “Ideal lattice and the structure of rings”, Trans. Am. Math. Soc., 75:1 (1953), 136–153 | DOI | MR | Zbl

[42] Boynton J., “Pullbacks of Arithmetical Rings”, Commun. Algebra., 35:9 (2007), 2671–2684 | DOI | MR | Zbl

[43] Boynton J., “Prüfer conditions and the total quotient ring”, Commun. Algebra., 39:5 (2011), 1624–1630 | DOI | MR | Zbl

[44] Burgess W. D., Stephenson W., “An analogue of the Pierce sheaf for non-commutative rings”, Commun. Algebra., 6:9 (1978), 863–886 | DOI | MR | Zbl

[45] Burgess W. D., Stephenson W., “Rings all of whose Pierce stalks are local”, Can. Math. Bull., 22:2 (1979), 159–164 | DOI | MR | Zbl

[46] Caballero C. E., “Self-duality and $\ell $-hereditary semidistributive rings”, Commun. Algebra., 14:10 (1986), 1821–1843 | DOI | MR | Zbl

[47] Camillo V., “Distributive modules”, J. Algebra., 36:1 (1975), 16–25 | DOI | MR | Zbl

[48] Chatters A. W., “A decomposition theorem for Noetherian hereditary rings”, Bull. London Math. Soc., 4 (1972), 125–126 | DOI | MR | Zbl

[49] Arithmetical Properties of Commutative Rings and Monoids, ed. Chapman S. T., CRC Press, 2005 | MR | Zbl

[50] Chouinard L. G., Hardy B. R., Shores T. S., “Arithmetical semihereditary semigroup rings”, Commun. Algebra., 8:17, 1593–1652 | DOI | MR | Zbl

[51] Couchot F., “Localization of injective modules over arithmetical rings”, Commun. Algebra., 37:10 (2009), 3418–3423 | DOI | MR | Zbl

[52] Couchot F., “Finitistic weak dimension of commutative arithmetical rings”, Arab. J. Math., 1:1 (2012), 63–67 | DOI | MR | Zbl

[53] Couchot F., “Gaussian trivial ring extensions and FQP-rings”, Commun. Algebra., 43:7 (2015), 2863–2874 | DOI | MR | Zbl

[54] Dung N. V., Huynh D. V., Smith P. F., Wisbauer R., Extending Modules, Wiley, New York, 1994 | MR

[55] Eisenbud D., Griffith Ph., “Serial rings”, J. Algebra., 17 (1971), 389–400 | DOI | MR | Zbl

[56] Er N., “Rings whose modules are direct sums of extending modules”, Proc. Am. Math. Soc., 137:7 (2009), 2265–2271 | DOI | MR | Zbl

[57] Faith C., Algebra: Rings, Modules and Categories I, Springer-Verlag, Berlin–New York, 1973 | MR | Zbl

[58] Faith C., Algebra II, Springer-Verlag, Berlin–New York, 1976 | MR | Zbl

[59] Fuchs L., “Über die Ideale arithmetischer Ringe”, Comment. Math. Helvet., 23:1 (1949), 334–341 | DOI | MR | Zbl

[60] Fuchs L., Heinzer W., Olberding B., “Commutative ideal theory without finiteness conditions: Primal ideals”, Trans. Am. Math. Soc., 357 (2005), 2771–2798 | DOI | MR | Zbl

[61] Fuchs L., Heinzer W., Olberding B., “Commutative ideal theory without finiteness conditions: Completely irreducible ideals”, Trans. Am. Math. Soc., 358:7 (2006), 3113–3131 | DOI | MR | Zbl

[62] Gillman L., Henriksen M., “Rings of continuous functions in which every finitely generated ideal is principal”, Trans. Am. Math. Soc., 82:2 (1956), 366–391 | DOI | MR | Zbl

[63] Gillman L., Henriksen M., “Some remarks about elementary divisor rings”, Trans. Am. Math. Soc., 82:2 (1956), 362–365 | DOI | MR | Zbl

[64] Glaz S., Schwarz R., “Prüfer conditions in commutative rings”, Arab. J. Sci. Engin., 36 (2011), 967–983 | DOI | MR

[65] Goodearl K. R., Ring Theory, Marcel Dekker, New York, 1976 | MR | Zbl

[66] Goodearl K. R., Von Neumann Regular Rings, Pitman, London, 1979 | MR | Zbl

[67] Goodearl K. R., Warfield R. B., An Introduction to Noncommutative Noetherian Rings, Cambridge Univ. Press, Cambridge, 1989 | MR | Zbl

[68] Gordon R., Robson J. C., “Krull dimension”, Mem. Am. Math. Soc., 1973, no. 133, 1–78 | MR

[69] Gräter G., General Lattice Theory. Second edition, Burkhäuser, Basel–Boston–Berlin, 2003 | MR

[70] Gräter J., “Über die Distributivität des Idealverbandes eines kommutativen Ringes”, Monatsh. Math., 99:4 (1985), 267–278 | DOI | MR | Zbl

[71] Handelman D., “Strongly semiprime rings”, Pac. J. Math., 60:1 (1975), 115–122 | DOI | MR | Zbl

[72] Handelman D., Lawrence J., “Strongly prime rings”, Trans. Am. Math. Soc., 211 (1975), 209–223 | DOI | MR | Zbl

[73] Hardy B. R., Shores T. S., “Arithmetical semigroup rings”, Can. J. Math., 32:6 (1980), 1361–1371 | DOI | MR | Zbl

[74] Hattori A., “A foundation of torsion theory for modules over general rings”, Nagoya Math. J., 17 (1960), 147–158 | DOI | MR | Zbl

[75] Heinzer W. J., Ratliff L. J., Rush D. E., “Strongly irreducible ideals of a commutative ring”, J. Pure Appl. Algebra., 166:3 (2002), 267–275 | DOI | MR | Zbl

[76] Henriksen M., “Some remarks about elementary divisor rings, II”, Michigan Math. J., 3 (1956), 159–163 | MR | Zbl

[77] Hinohara Y., “Note on noncommutative local rings”, Nagoya Math. J., 17 (1960), 161–166 | DOI | MR | Zbl

[78] Jensen C. U., “A remark on arithmetical rings”, Proc. Am. Math. Soc., 15:6 (1964), 951–954 | DOI | MR | Zbl

[79] Jensen C. U., “Arithmetical rings”, Acta Math. Acad. Sci. Hung., 17:1–2 (1966), 115–123 | DOI | MR | Zbl

[80] Jeremy L., “Modules et anneaux quasi-continus”, Can. Math. Bull., 17:2 (1974), 217–228 | DOI | MR | Zbl

[81] Johnson R. E., Wong F. T., “Quasi-injective modules and irreducible rings”, J. London Math. Soc., 36 (1961), 260–268 | DOI | MR | Zbl

[82] Jondrup S., “p.p. rings and finitely generated flat ideals”, Trans. Am. Math. Soc., 28:2 (1971), 431–435 | MR

[83] Kaplansky I., “Elementary divisors and modules”, Trans. Am. Math. Soc., 19:2 (1949), 21–23 | MR

[84] Koike K., “Self-duality of quasi-Harada rings and locally distributive rings”, J. Algebra., 302:2 (2006), 613–645 | DOI | MR | Zbl

[85] Lam T. Y., Exercises in Classical Ring Theory, Springer-Verlag, New York, 1995 | MR

[86] Lam T. Y., Lectures on Modules and Rings, Springer-Verlag, New York, 1999 | MR | Zbl

[87] Larsen M. D., Lewis W. J., Shores T. S., “Elementary divisor rings and finitely presented modules”, Trans. Am. Math. Soc., 187:1 (1974), 231–248 | DOI | MR | Zbl

[88] Lawrence J., “A singular primitive ring”, Trans. Am. Math. Soc., 45:1 (1974), 59–62 | MR | Zbl

[89] Lee T. K., Zhou`Y., “Modules which are invariant under automorphisms of their injective hulls”, J. Algebra Appl., 6:2 (2013), 1250159 | DOI | MR

[90] Lemonnier B., “Sur les anneaux qui ont une déviation”, C. R. Acad. Sci. Paris. Ser. A., 275 (1972), 357–359 | MR | Zbl

[91] Lemonnier B., “Dimension de Krull et codéviation des anneaux semi-héréditaires”, C. R. Acad. Sci. Paris. Ser. A., 284 (1977), 663–666 | MR | Zbl

[92] Levy L. S., “Sometimes only square matrices can be diagonalized”, Proc. Am. Math. Soc., 52 (1975), 18–22 | DOI | MR | Zbl

[93] Lu X., “Some properties of completely arithmetical rings”, Algebra Colloq., 23:1 (2016), 83–88 | DOI | MR | Zbl

[94] Lu X., Boynton J., “Completely arithmetical rings”, Commun. Algebra., 42:9 (2014), 4047–4054 | DOI | MR | Zbl

[95] Lucas T. G., “Strong Prüfer rings and the ring of finite fractions”, J. Pure Appl. Algebra., 84 (1993), 59–71 | DOI | MR | Zbl

[96] Lucas T. G., “The Gaussian property for rings and polynomials”, Houston J. Math., 34:1 (2008), 1–18 | MR | Zbl

[97] McGovern W., Sharma M., “Gaussian Property of the Rings $R(X)$ and $R\langle X\rangle$”, Commun. Algebra., 44:4 (2016), 1636–1646 | DOI | MR | Zbl

[98] Menzel W., “Ein Kriterium für Distributivität des Untergruppenverbands einer Abelschen Operatorgruppe”, Math. Z., 75:3 (1961), 271–276 | DOI | MR | Zbl

[99] Michler G., Wille R., “Die primitiven Klassen arithmetischer Ringe”, Math. Z., 113:5 (1970), 369–372 | DOI | MR | Zbl

[100] Min L. S., “On the constructions of local and arithmetical rings”, Acta. Math. Acad. Sci. Hung., 32:1–2 (1978), 31–34 | DOI | MR | Zbl

[101] Nicholson W. K., “Lifting idempotents and exchange rings”, Trans. Am. Math. Soc., 229:2 (1977), 269–278 | DOI | MR | Zbl

[102] Nikseresht A., Aziz A., “On arithmetical rings and the radical formula”, Vietnam J. Math., 38:1 (2010), 55–62 | MR | Zbl

[103] van Oystaeyen F., “Generalized Rees rings and arithmetical graded rings”, J. Algebra., 82:1 (1983), 185–193 | DOI | MR | Zbl

[104] Parkash A., “Arithmetical rings satisfy the radical formula”, J. Commun. Algebra., 4:2 (2012), 293–296 | DOI | MR | Zbl

[105] Puninski G. E., “Projective modules over the endomorphism ring of a biuniform module”, J. Pure Appl. Algebra., 188:1–3 (2004), 227–246 | DOI | MR | Zbl

[106] Rowen L. H., Ring Theory, Academic Press, Boston, 1988 | MR

[107] Shores T., Lewis W. J., “Serial modules and endomorphism rings”, Duke Math. J., 41:4 (1974), 889–909 | DOI | MR | Zbl

[108] Stenström B., Rings of Quotients, Springer-Verlag, Berlin–New York, 1975 | MR | Zbl

[109] Stephenson W., “Modules whose lattice of submodules is distributive”, Proc. London Math. Soc., 28:2 (1974), 291–310 | DOI | MR | Zbl

[110] Tuganbaev A. A., Semidistributive Modules and Rings, Kluwer Academic Publ., Dordrecht, Boston, London, 1998 | MR | Zbl

[111] Tuganbaev A. A., Rings Close to Regular, Kluwer Academic Publ., Dordrecht, Boston, London, 2002 | MR | Zbl

[112] Tuganbaev A. A., “Multiplication modules”, J. Math. Sci., 123:2 (2004), 3839–3905 | DOI | MR | Zbl

[113] Warfield R. B., “Decomposability of finitely presented modules”, Proc. Am. Math. Soc., 25 (1970), 167–172 | DOI | MR | Zbl

[114] Warfield R. B., “Stable generation of modules”, Lect. Notes Math., 700 (1979), 16–33 | DOI | MR | Zbl

[115] Wright M. H., “Right locally distributive rings”, Ring Theory, Proc. Bien. Ohio State (Denison Conf., Granville, Ohio, May, 1992), Singapore, 1993, 350–357 | MR | Zbl

[116] Xue W., “Rings with Morita duality”, Lect. Notes Math., 1523 (1992), 1–197 | DOI | MR

[117] Yaremenko Yu., “Noetherian semi-perfect rings of distributive module type”, Mat. Stud., 8:1 (1997), 3–10 | MR | Zbl

[118] Yukimoto Y., “Artinian rings whose projective indecomposables are distributive”, Osaka J. Math., 22 (1985), 339–344 | MR | Zbl

[119] Wisbauer R., Foundations of Module and Ring Theory, Gordon Breach, Philadelphia, 1991 | MR | Zbl