Geodesic
Rings of Laurent series, Laurent rings, and Malcev--Neumann rings
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Algebra, Tome 184 (2020), pp. 3-111.

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This paper is a review with proofs of ring-theoretical properties of rings of skew Laurent series $A((x,\varphi))$ over a ring $A$, where $A$ is an associative ring with nonzero identity element. In addition, we consider Laurent rings and Malcev–Neumann rings which are proper extensions of skew Laurent series rings.
Keywords: ring of skew Laurent series, Laurent ring, Malcev–Neumann ring. 
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     author = {A. A. Tuganbaev},
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}
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A. A. Tuganbaev. Rings of Laurent series, Laurent rings, and Malcev--Neumann rings. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Algebra, Tome 184 (2020), pp. 3-111. http://geodesic.mathdoc.fr/item/INTO_2020_184_a0/

[1] Gelfand I. M., Dikii L. A., “Integriruemye nelineinye uravneniya i teorema Liuvillya”, Funkts. anal. prilozh., 13:1 (1979), 8–20 | MR | Zbl

[2] \label{[21]} Dzhumadildaev A. S., “Differentsirovaniya i tsentralnye rasshireniya algebry Li formalnykh psevdodifferentsialnykh operatorov”, Algebra i analiz., 6:1 (1994), 140–158

[3] Parshin A. N., “O koltse formalnykh psevdodifferentsialnykh operatorov”, Tr. Mat. in-ta im. V. A. Steklova RAN., 224 (1999), 291–305 | Zbl

[4] Sonin K. I., “Regulyarnye koltsa ryadov Lorana”, Fundam. prikl. mat., 1:1 (1995), 315–317 | MR | Zbl

[5] Sonin K. I., “Regulyarnye koltsa kosykh ryadov Lorana”, Fundam. prikl. mat., 1:2 (1995), 565–568 | MR | Zbl

[6] Sonin K. I., “Biregulyarnye koltsa ryadov Lorana”, Vestn. Mosk. un-ta. Ser. 1. Mat. mekh., 1997, no. 4, 20–22 | MR | Zbl

[7] Tuganbaev A. A., “Koltsa kosykh ryadov Lorana i uslovie maksimalnosti dlya pravykh annulyatorov”, Diskr. mat., 20:1 (2008), 80–86 | Zbl

[8] Tuganbaev A. A., “O polutsepnykh koltsakh”, Diskr. mat., 28:4 (2016), 150–157 | MR

[9] Tuganbaev D. A., “Tsepnye koltsa ryadov Lorana”, Fundam. pri. mat., 3:3 (1997), 947–951 | MR | Zbl

[10] Tuganbaev D. A., “Tsepnye koltsa kosykh ryadov Lorana”, Vestn. Mosk. un-ta. Ser. 1. Mat. mekh., 2000, no. 1, 52–55 | Zbl

[11] Tuganbaev D. A., “Koltsa kosykh ryadov Lorana i koltsa glavnykh idealov”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 2000, no. 5, 55–57 | Zbl

[12] Tuganbaev D. A., “Polulokalnye distributivnye koltsa kosykh ryadov Lorana”, Fundam. prikl. mat., 6:3 (2000), 913–921 | MR | Zbl

[13] Tuganbaev D. A., “Koltsa psevdodifferentsialnykh operatorov i usloviya na tsepi”, Vestn. Mosk. un-ta. Ser. 1. Mat. mekh., 2002, no. 4, 26–32 | Zbl

[14] Tuganbaev D. A., “Maltsevskie koltsa”, Diskr. mat., 20:2 (2008), 63–81 | Zbl

[15] Alhevaz A., Kiani D., “Radicals of skew inverse Laurent series rings”, Commun. Algebra., 41:8 (2013), 2884–2902 | DOI | MR | Zbl

[16] Alhevaz A., Kiani D., “On zero divisors in skew inverse Laurent series over noncommutative rings”, Commun. Algebra., 42:2 (2014), 469–487 | DOI | MR | Zbl

[17] \label{[3]} Amitsur S. A., “Radicals of polynomial rings”, Can. J. Math., 8 (1956), 355–361 | DOI | MR | Zbl

[18] Bergman G. M., “Conjugates and $n$th roots in Hahn–Laurent group rings”, Bull. Malaysian Math. Soc., 1:2 (1978), 29–41 ; “Historical addendum”, Bull. Malaysian Math. Soc., 2:2 (1979), 41–42 | MR | Zbl | MR | Zbl

[19] Cohn P. M., Skew Fields, Cambridge Univ. Press, Cambridge, 1995 | MR | Zbl

[20] Elliott G. A., Ribenboim P., “Fields of generalized power series”, Arch. Math., 54:4 (1990), 365–371 | DOI | MR | Zbl

[21] Facchini A., Semilocal Categories and Modules with Semilocal Endomorphism Rings, Birkhäuser, Basel, 2019 | MR | Zbl

[22] Facchini A., Herbera D., “Local morphisms and modules with a semilocal endomorphism ring”, Alg. Represent. Th., 9:4 (2006), 403–422 | DOI | MR | Zbl

[23] Faith C., Algebra: Rings, Modules, and Categories, Springer-Verlag, Berlin, 1973 | MR | Zbl

[24] \label{[24]} Faith C., Algebra: Ring Theory, Springer-Verlag, New York, 1976 | MR

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

[26] Goodearl K. R., “Centralizers in differential, pseudo-differential, and fractional differential operator rings”, Rocky Mountain J. Math., 13:4 (1983), 573–618 | DOI | MR | Zbl

[27] Goodearl K. R., Small L. W., “Krull versus global dimension in Noetherian PI-rings”, Proc. Am. Math. Soc., 92:2 (1984), 175–178 | DOI | MR | Zbl

[28] \label{[5]} Goodearl K. R., Warfield R. B., An Introduction to Noncommutative Noetherian Rings, Cambridge Univ. Press, Cambridge, 2004 | MR | Zbl

[29] Guillemin V., Quillen D., Sternberg S., “The integrability of characteristics”, Commun. Pure Appl. Math., 23 (1970), 39–77 | DOI | MR | Zbl

[30] Guillemin V., Sternberg S., Symplectic Techniques in Physics, Cambridge Univ. Press, Cambridge, 1984 | MR | Zbl

[31] Johns B., “Chain conditions and nil ideals”, J. Algebra., 73:2 (1981), 287–294 | DOI | MR | Zbl

[32] Hashemi E., Hamidizadeh M., Alhevaz A., “On the Jacobson radical of skew inverse Laurent series ring”, J. Algebra Appl., 18:9 (2019), 1950177.1–50177 | DOI | MR

[33] \label{[6]} Karimi S., Sahebi Sh., Habibi M., “The Jacobson radical of inverse Laurent power series of rings”, J. Algebra Appl., 17:4 (2018), 1850061.1–1850061.6 | DOI | MR

[34] \label{[49]} Lam T. Y., Lectures on Modules and Rings, Springer-Verlag, New York, 1999 | MR | Zbl

[35] \label{[52]} Lanski C., “Nil subrings of Goldie rings are nilpotent”, Can. J. Math., 21 (1969), 904–907 | DOI | MR | Zbl

[36] Liu Z., “On $n$-root closedness of generalized power series rings over pairs of rings”, J. Pure Appl. Algebra., 144:3 (1999), 303–312 | DOI | MR | Zbl

[37] Liu Z., “A note on Hopfian modules”, Commun. Algebra., 28:6 (2000), 3031–3040 | DOI | MR | Zbl

[38] Liu Z., “The ascending chain condition for principal ideals of rings of generalized power series”, Commun. Algebra., 32:9 (2004), 3305–3314 | DOI | MR | Zbl

[39] Liu Z., Ahsan J., “The Tor-groups of modules of generalized power series”, Algebra Colloq., 12 (2005), 477–484 | DOI | MR | Zbl

[40] Liu Z., Cheng H., “Quasi-duality for the rings of generalized power series”, Commun. Algebra., 28:3 (2000), 1175–1188 | DOI | MR | Zbl

[41] Liu Z., Li F., “PS-rings of generalized power series”, Commun. Algebra., 26:7 (1998), 2283–2291 | DOI | MR | Zbl

[42] Lorenz M., “Division algebras generated by finitely generated nilpotent groups”, J. Algebra., 85 (1983), 368–381 | DOI | MR | Zbl

[43] Makar-Limanov L., “The skew field of fractions of the first Weyl algebra contains a free noncommutative subalgebra”, Commun. Algebra., 11:17 (1983), 2003–2006 | DOI | MR | Zbl

[44] Mal'cev A. I., “On the embedding of group algebras in division algebras”, Dokl. Akad. Nauk SSSR., 60 (1948), 1499–1501 | MR

[45] \label{[59]} Manaviyat R., Moussavi A., “On annihilator ideals of pseudo-differential operator rings”, Algebra Colloq., 4 (22) (2015), 607–620 | DOI | MR | Zbl

[46] Mulase M., “Solvability of the super KP equation and a generalization of the Birkhoff decomposition”, Invent. Math., 92 (1988), 1–46 | DOI | MR | Zbl

[47] Musson I., Stafford K., “Malcev–Neumann group rings”, Commun. Algebra., 21:6 (1993), 2065–2075 | DOI | MR | Zbl

[48] Neumann B. H., “On ordered division rings”, Trans. Am. Math. Soc., 66 (1949), 202–252 | DOI | MR | Zbl

[49] Paykan K., Moussavi A., “Special properties of differential inverse power series rings”, J. Algebra Appl., 15:9 (2016), 1650181.1–1650181.23 | DOI | MR

[50] Paykan K., Moussavi A., “Study of skew inverse Laurent series rings”, J. Algebra Appl., 16:11 (2017), 1750221.1–1750221.33 | DOI | MR

[51] Paykan K., Moussavi A., “Primitivity of skew inverse Laurent series rings and related rings”, J. Algebra Appl., 18:6 (2019), 1950116.1–1950116.12 | DOI | MR

[52] Ribenboim P., “Squares in fields of generalized power series”, C. R. Math. Rep. Acad. Sci. Can., 12:5 (1990), 199–203 | MR | Zbl

[53] Ribenboim P., “Rings of generalized power series: nilpotent elements”, Abh. Math. Sem. Univ. Hamburg., 61 (1991), 15–33 | DOI | MR | Zbl

[54] Ribenboim P., “Noetherian rings of generalized power series”, J. Pure Appl. Algebra., 79:3 (1992), 293–312 | DOI | MR | Zbl

[55] Ribenboim P., “Extension of Hironaka's standard basis theorem for generalized power series”, Arch. Math., 60:5 (1993), 436–439 | DOI | MR | Zbl

[56] Ribenboim P., “Multiplicative structures on power series and the construction of skewfields”, C. R. Acad. Sci. Paris. Ser. I. Math., 316:11 (1993), 1117–1121 | MR | Zbl

[57] Ribenboim P., “Rings of generalized power series. II. Units and zero-divisors”, J. Algebra., 168:1 (1994), 71–89 | DOI | MR | Zbl

[58] Ribenboim P., “Special properties of generalized power series”, J. Algebra., 173:3 (1995), 566–586 | DOI | MR | Zbl

[59] Ribenboim P., “Multiplicative structures on power series and the construction of skewfields”, J. Algebra., 185:2 (1996), 267–297 | DOI | MR | Zbl

[60] Ribenboim P., “Semisimple rings and von Neumann regular rings of generalized power series”, J. Algebra., 198:2 (1997), 327–338 | DOI | MR | Zbl

[61] Risman L., “Twisted rational functions and series”, J. Pure Appl. Algebra., 12 (1978), 181–199 | DOI | MR | Zbl

[62] \label{[76]} Rowen L. H., Ring Theory, Academic Press, New York, 1988 | MR

[63] \label{13} Sato M., Sato Y., “Soliton equations as dynamical systems on infinite dimensional Grassman manifold”, Lect. Notes Num. Appl. Anal., 5 (1982), 259–271 | MR

[64] Schur I., “Über vertauschbare lineare Differentialausdrücke”, Sitzungsber. Berliner Math. Ges., 4 (1905), 2–8 | Zbl

[65] Shock R. C., “Nil subrings in finiteness conditions”, Am. Math. Mon., 78 (1971), 741–748 | DOI | MR | Zbl

[66] Smits T. H., “Skew-Laurent series over semisimple rings”, Delft. Progr. Rep., 2 (1977), 131–136 | MR | Zbl

[67] Sonin K., “Krull dimension of Malcev–Neumann rings”, Commun. Algebra., 26:9 (1998), 2915–2931 | DOI | MR | Zbl

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

[69] Tuganbaev A. A., Semidistributive Modules and Rings, Kluwer, Dordrecht–Boston–London, 1998 | MR | Zbl

[70] Tuganbaev A. A., Distributive Modules and Related Topics, Gordon Breach, Amsterdam, 1999 | MR | Zbl

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

[72] Tuganbaev A. A., “Polynomial and series rings and principal ideals”, J. Math. Sci., 114:2 (2003), 1204–1226 | DOI | MR | Zbl

[73] Tuganbaev A. A., “The Jacobson radical of the Laurent series ring”, J. Math. Sci., 149:2 (2008), 1182–1186 | DOI | MR | Zbl

[74] Tuganbaev A. A., Semidistributive Laurent Series Rings, arXiv: 2006.06757 [math.RA] | MR

[75] Tuganbaev A. A., “Right serial skew Laurent series rings”, J. Algebra Appl., 2021 | Zbl

[76] Tuganbaev D. A., “Some ring and module properties of skew power series”, Formal Power Series and Algebraic Combinatorics, eds. Krob D., Mikhalev A. A., Mikhalev A. V., Springer-Verlag, Berlin–Heidelberg–New York, 2000, 613–622 | DOI | MR | Zbl

[77] Tuganbaev D. A., “Laurent series rings and pseudo-differential operator rings”, J. Math. Sci., 128:3 (2005), 2843–2893 | DOI | MR | Zbl

[78] Tuganbaev D. A., “Jacobson radical and Laurent series rings”, J. Math. Sci., 152:2 (2008), 304–306 | DOI | MR | Zbl

[79] Tuganbaev D. A., “Laurent rings”, J. Math. Sci., 149:3 (2008), 1286–1337 | DOI | MR | Zbl

[80] Zhang H., “On semilocal rings”, Proc. Am. Math. Soc., 137 (2009), 845–852 | DOI | MR | Zbl

[81] Ziembowski M., “Laurent serial rings over a semiperfect ring can not be semiperfect”, Commun. Algebra., 42 (2014), 664–666 | DOI | MR | Zbl