Around Baer--Kaplansky theorem

P. A. Krylov ; A. A. Tuganbaev ; A. V. Tsarev

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Algebra, Tome 159 (2019), pp. 46-67.

Résumé

Using the example of modules and a number of familiar Abelian groups, we demonstrate the Kaplansky method of proving isomorphism theorems for endomorphism rings.

Détail

Keywords: Abelian group, endomorphism ring, isomorphism theorem for endomorphism rings, Baer–Kaplansky theorem, Kaplansky method.

Bibliographie

[1] Krylov P. A., “Silno odnorodnye abelevy gruppy bez krucheniya”, Sib. mat. zh., 24:2 (1983), 77–84 | MR | Zbl

[2] Krylov P. A., Mikhalev A. V., Tuganbaev A. A., Abelevy gruppy i ikh koltsa endomorfizmov, Faktorial Press, M., 2006

[3] Krylov P. A., Tuganbaev A. A., Moduli nad oblastyami diskretnogo normirovaniya, Faktorial Press, M., 2007

[4] Krylov P. A., Tuganbaev A. A., “Moduli nad koltsami formalnykh matrits”, Fundam. prikl. mat., 15:8 (2009), 145–211

[5] Krylov P. A., Tuganbaev A. A., “Formalnye matritsy i ikh opredeliteli”, Fundam. prikl. mat., 19:1 (2014), 65–119

[6] Krylov P. A., Tuganbaev A. A., Koltsa formalnykh matrits i moduli nad nimi, MTsNMO, M., 2017 | MR

[7] Fuks L., Beskonechnye abelevy gruppy, v. 1, Mir, M., 1973

[8] Baer R., “Automorphism rings of primary Abelian operator groups”, Ann. Math., 44:2 (1943), 192–227 | DOI | MR | Zbl

[9] Baer R., Linear Algebra and Projective Geometry, Academic Press, New York, 1952 | MR | Zbl

[10] Bazzoni S., Metelli C., “On Abelian torsion-free separable groups and their endomorphism rings”, Symp. Math., 23 (1979), 259–285 | MR | Zbl

[11] Files S. T., “Endomorphisms of local Warfield groups”, Contemp. Math., 171 (1994), 99–107 | DOI | MR | Zbl

[12] Files S. T., “Outer automorphisms of endomorphism rings of Warfield groups”, Arch. Math., 65 (1995), 15–22 | DOI | MR | Zbl

[13] Files S. T., “Endomorphism algebras of modules with distinguished torsion-free elements”, J. Algebra, 178 (1995), 264–276 | DOI | MR | Zbl

[14] Flagg M., “A Jacobson radical isomorphism theorem for torsion-free modules”, Models, Modules and Abelian Groups, eds. Göbel R., Goldsmith B., Walter de Gruyter, Berlin, 2008, 309–314 | MR | Zbl

[15] Flagg M., “Jacobson radical isomorphism theorems for mixed modules. Part one: determining the torsion”, Commun. Algebra, 37:5 (2009), 1739–1747 | DOI | MR | Zbl

[16] Flagg M., “The role of the Jacobson radical in isomorphism theorems”, Contemp. Math., 576 (2012), 77–88 | DOI | MR | Zbl

[17] Flagg M., “The Jacobson radical's role in isomorphism theorems for $p$-adic modules extends to topogical isomorphism”, Groups, Modules, and Model Theory—Surveys and Recent Developments, eds. Droste M., Fuchs L., Goldsmith B., Strüngmann L., Springer, 2017, 285–300 | DOI | MR

[18] Göbel R., May W., “The construction of mixed modules from torsion modules”, Arch. Math., 48 (1987), 476–490 | DOI | MR | Zbl

[19] Hausen J., Johnson J. A., “Determining Abelian $p$-groups by the Jacobson radical of their endomorphism rings”, J. Algebra, 174:1 (1995), 217–224 | DOI | MR | Zbl

[20] Hausen J., Praeger C. E., Schultz P., “Most Abelian $p$-groups are determined by the Jacobson radical of their endomorphism rings”, Math. Z., 216 (1994), 431–436 | DOI | MR | Zbl

[21] Kaplansky I., Infinite Abelian Groups, Univ. of Michigan Press, Ann Arbor, 1954 | MR | Zbl

[22] May W., “Endomorphism rings of Abelian groups with ample divisible subgroups”, Bull. London Math. Soc., 10:3 (1978), 270–272 | DOI | MR | Zbl

[23] May W., “Endomorphism rings of mixed Abelian groups”, Contemp. Math., 87 (1989), 61–74 | DOI | MR | Zbl

[24] May W., “Isomorphisms of endomorphism algebras over complete discrete valuation rings”, Math. Z., 204:4 (1990), 485–499 | DOI | MR | Zbl

[25] May W., “Endomorphism algebras of not necessarily cotorsion-free modules”, Abelian Groups and Noncommutative Rings, eds. Fuchs L., Goodearl K. R., Stafford J. T., Vinsonhaler C., Am. Math. Soc., Providence, Rhode Island, 1992, 257–264 | MR

[26] May W., “Endomorphisms over incomplete discrete valuation domains”, Contemp. Math., 171 (1994), 277–285 | DOI | MR | Zbl

[27] May W., “The theorem of Baer and Kaplansky for mixed modules”, J. Algebra, 177 (1995), 255–263 | DOI | MR | Zbl

[28] May W., “The use of the finite topology on endomorphism rings”, J. Pure Appl. Algebra, 163 (2001), 107–117 | DOI | MR | Zbl

[29] May W., Toubassi E., “Endomorphisms of Abelian groups and the theorem of Baer and Kaplansky”, J. Algebra, 43:1 (1976), 1–13 | DOI | MR | Zbl

[30] May W., Toubassi E., “A result on problem 87 of L. Fuchs”, Lect. Notes Math., 616, 1977, 354–367 | DOI | MR | Zbl

[31] May W., Toubassi E., “Isomorphisms of endomorphism rings of rank one mixed groups”, J. Algebra, 71:2 (1981), 508–514 | DOI | MR | Zbl

[32] May W., Toubassi E., “Endomorphisms of rank one mixed modules over discrete valuation rings”, Pac. J. Math., 108:1 (1983), 155–163 | DOI | MR | Zbl

[33] Mikhalev A. V., “Isomorphisms and anti-isomorphisms of endomorphism rings of modules”, Proc. First Int. Algebraic Workshop, Berlin, 1996, 69–122 | MR | Zbl

[34] Schultz P., “When is an Abelian $p$-group determined by the Jacobson radical of its endomorphism ring”, Contemp. Math., 171, 1994, 385–396 | DOI | MR | Zbl

[35] Toubassi E., May W., “Classifying endomorphism rings of rank one mixed groups”, Abelian Groups and Modules, Proc. Conf. (Udine, 1984), Springer, Vienna, 1984, 253–263 | MR