sp-Groups and their endomorphism rings

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. 68-110.

Résumé

$sp$-Groups form an interesting and informative class of Abelian mixed groups. In this paper, we systematically study self-small $sp$-groups of finite rank and their endomorphism rings.

Détail

Keywords: Abelian $sp$-group, self-small group, endomorphism ring, Walker category.

Bibliographie

[1] Zinovev E. G., Koltsa psevdoalgebraicheskikh chisel i moduli nad nimi, Diss. na soisk. uch. step. kand. fiz.-mat. nauk, M., 2009

[2] Krylov P. A., “Summy avtomorfizmov abelevykh grupp i radikal Dzhekobsona koltsa endomorfizmov”, Izv. vuzov. Mat., 4 (1976), 56–66 | Zbl

[3] Krylov P. A., “Smeshannye abelevy gruppy kak moduli nad svoimi koltsami endomorfizmov”, Fundam. prikl. mat., 6:3 (2000), 793–812 | MR | Zbl

[4] Krylov P. A., “Nasledstvennye koltsa endomorfizmov smeshannykh abelevykh grupp”, Sib. mat. zh., 43:1 (2002), 108–119 | MR | Zbl

[5] Krylov P. A., “Radikal Dzhekobsona koltsa endomorfizmov abelevoi gruppy”, Algebra i logika, 43:1 (2004), 60–76 | MR | Zbl

[6] Krylov P. A., Pakhomova E. G., “Abelevy gruppy i regulyarnye moduli”, Mat. zametki, 69:3 (2001), 402–411 | DOI | MR | Zbl

[7] Sorokin K. S., Abelevy gruppy s chistymi koltsami endomorfizmov, Diss. na soisk. uch. step. kand. fiz.-mat. nauk, Tomsk, 2014

[8] Timoshenko E. A., “O bazovykh polyakh $csp$-kolets”, Algebra i logika, 49:4 (2010), 555–565 | MR | Zbl

[9] Timoshenko E. A., “Proektivnye moduli nad koltsom psevdoratsionalnykh chisel”, Zh. SFU. Ser. mat. fiz., 4:4 (2011), 541–550

[10] Timoshenko E. A., “Proektivnye moduli nad csp-koltsami”, Zh. SFU. Ser. mat. fiz., 5:4 (2012), 581–585

[11] Timoshenko E. A., “Chisto transtsendentnye rasshireniya polya ratsionalnykh chisel kak bazovye polya $csp$-kolets”, Vestn. Tomsk. gos. un-ta. Mat. mekh., 2013, no. 5 (25), 30–39

[12] Timoshenko E. A., “O bazovykh polyakh $csp$-kolets, II”, Fundam. prikl. mat., 20:5 (2015), 149–156

[13] Tuganbaev A. A., Teoriya kolets. Arifmeticheskie moduli i koltsa, MTsNMO, M., 2009

[14] Fomin A. A., Chislovye koltsa i moduli nad nimi, Prometei, M., 2013

[15] Tsarev A. V., “Psevdoratsionalnyi rang abelevoi gruppy”, Sib. mat. zh., 46:1 (2005), 217–229 | MR | Zbl

[16] Tsarev A. V., “Proektivnye i obrazuyuschie moduli nad koltsom psevdoratsionalnykh chisel”, Mat. zametki, 80:3 (2006), 437–448 | DOI | MR | Zbl

[17] Tsarev A. V., “Nekotorye morfizmy modulei nad koltsom psevdoratsionalnykh chisel”, Sib. mat. zh., 49:4 (2008), 945–953 | MR | Zbl

[18] Tsarev A. V., “Servantnye podkoltsa kolets $\mathbb{Z}_\chi$”, Mat. sb., 200:10 (2009), 123–150 | DOI | Zbl

[19] Tsarev A. V., Moduli nad koltsom psevdoratsionalnykh chisel i faktorno delimye gruppy, Diss. na soisk. uch. step. dokt. fiz.-mat. nauk, M., 2009

[20] Albrecht U. F., “Mixed Abelian groups with Artinian quasi-endomorphism ring”, Commun. Algebra, 25:11 (1997), 497–511 | DOI | MR

[21] Albrecht U. F., “A-projective resolutions and an Azumaya theorem for a class of mixed abelian groups”, Czechoslovak. Math. J., 51:1 (2001), 73–93 | DOI | MR | Zbl

[22] Albrecht U. F., Breaz S., Vinsonhaler C. Wickless W., “Cancellation properties for quotient divisible groups”, J. Algebra, 317:1 (2007), 424–434 | DOI | MR | Zbl

[23] Albrecht U. F., Breaz S., Wickless W., “Purity and self-small groups”, Commun. Algebra, 35:11 (2007), 3789–3807 | DOI | MR | Zbl

[24] Albrecht U. F., Breaz S., Wickless W., “Self-small abelian groups”, Bull. Austr. Math. Soc., 80:2 (2009), 205–216 | DOI | MR | Zbl

[25] Albrecht U. F., Goeters H. P., Wickless W., “The flat dimension of mixed abelian groups as $E$-modules”, Rocky Mount. J. Math., 25:2 (1995), 569–590 | DOI | MR | Zbl

[26] Albrecht U. F., Wickless W., “Homological properties of quotient divisible Abelian groups”, Commun. Algebra, 32:6 (2004), 2407–2423 | DOI | MR | Zbl

[27] Bartoszyński T., Judah H., Set Theory. On the Structure of the Real Line, A. K. Peters, Wellesley, MA, 1995 | MR | Zbl

[28] Blass A., “Combinatorial cardinal characteristics of the continuum”, Handbook of Set Theory, v. 1, eds. Foreman M., Kanamori A., Springer, Dordrecht, 2010, 395–489 | DOI | MR | Zbl

[29] Borooah G., Diesl A. J., Dorsey T. J., “Strongly clean triangular matrix rings over local rings”, J. Algebra, 312:2 (2007), 773–797 | DOI | MR | Zbl

[30] Borooah G., Diesl A. J., Dorsey T. J., “Strongly clean matrix rings over commutative local rings”, J. Pure Appl. Algebra, 212:1 (2008), 281–296 | DOI | MR | Zbl

[31] Bowshell R. A., Schultz P., “Unital rings whose additive endomorphisms commute”, Math. Ann., 228:3 (1977), 197–214 | DOI | MR | Zbl

[32] Breaz S., “On a class of mixed groups with semi-local Walk-endomorphism ring”, Commun. Algebra, 30:9 (2002), 4473–4485 | DOI | MR | Zbl

[33] Breaz S., “Self-small Abelian groups as modules over their endomorphism rings”, Commun. Algebra, 31:10 (2003), 4911–4924 | DOI | MR | Zbl

[34] Breaz S., “Quasi-decompositions for self-small abelian groups”, Commun. Algebra, 32:4 (2004), 1373–1384 | DOI | MR | Zbl

[35] Breaz S., “Warfield dualities induced by self-small mixed groups”, J. Group Theory, 13:3 (2010), 391–409 | DOI | MR | Zbl

[36] Breaz S., Schultz P., “Dualities for self-small groups”, Proc. Am. Math. Soc., 140:1 (2012), 69–82 | DOI | MR | Zbl

[37] Camillo V. P., Khurana D., “A characterization of unit regular rings”, Commun. Algebra, 29:5 (2001), 2293–2295 | DOI | MR | Zbl

[38] Camillo V. P., Khurana D., Lam T. Y., Nicholson W. K., Zhou Y., “Continuous modules are clean”, J. Algebra, 304:1 (2006), 94–111 | DOI | MR | Zbl

[39] Camillo V. P., Yu H. P., “Exchange rings, units and idempotents”, Commun. Algebra, 22:12 (1994), 4737–4749 | DOI | MR | Zbl

[40] Files S. T., Wickless W., “The Baer—Kaplansky theorem for a class of global mixed groups”, Rocky Mount. J. Math., 26:2 (1996), 593–613 | DOI | MR | Zbl

[41] Files S. T., Wickless W., “Direct sums of self-small mixed groups”, J. Algebra, 222:1 (1999), 1–16 | DOI | MR | Zbl

[42] Fomin A. A., “Some mixed abelian groups as modules over the ring of pseudo-rational numbers”, Abelian Groups and Modules, Birkhäuser, Basel, 1999, 87–100 | DOI | MR | Zbl

[43] Fomin A. A., “Quotient divisible mixed groups”, Contemp. Math., 273, 2001, 117–128 | DOI | MR | Zbl

[44] Fomin A. A., Wickless W., “Categories of mixed and torsion-free finite rank Abelian groups”, Abelian Groups and Modules, Springer, Dordrecht, 1995, 185–192 | DOI | MR

[45] Fomin A. A., Wickless W., “Self-small mixed Abelian groups $G$ with $G/T(G)$ finite rank divisible”, Commun. Algebra, 26:11 (1998), 3563–3580 | DOI | MR | Zbl

[46] Fomin A. A., Wickless W., “Quotient divisible Abelian groups”, Proc. Am. Math. Soc., 126:1 (1998), 45–52 | DOI | MR | Zbl

[47] Fuchs L., “Recent results and problems on abelian groups”, Topics in Abelian Groups, Chicago, 1963, 9–40 | MR

[48] Fuchs L., Rangaswamy K. M., “On generalized regular rings”, Math. Z., 107:1 (1968), 71–81 | DOI | MR | Zbl

[49] Glaz S., Wickless W., “Regular and principal projective endomorphism rings of mixed abelian groups”, Commun. Algebra, 22:4 (1994), 1161–1176 | DOI | MR | Zbl

[50] Göbel R., Opdenhövel A., “Every endomorphism of a local Warfield module of finite torsion-free rank is the sum of two automorphisms”, J. Algebra, 233:2 (2000), 758–771 | DOI | MR | Zbl

[51] Göbel R., Trlifaj J., Approximations and Endomorphism Algebras of Modules, Walter de Gruyter, Berlin, 2012 | MR | Zbl

[52] Goldsmith B., Pabst S., Scott A., “Unit sum number of rings and modules”, Quart. J. Math., 49:3 (1998), 331–344 | DOI | MR | Zbl

[53] Goldsmith B., Vámos P., “A note on clean Abelian groups”, Rend. Semin. Mat. Univ. Padova, 117 (2007), 181–191 | MR | Zbl

[54] Han J., Nicholson W. K., “Extensions of clean rings”, Commun. Algebra, 29:6 (2001), 2589–2595 | DOI | MR | Zbl

[55] Handelman D., “Perspectivity and cancellation in regular rings”, J. Algebra, 48:1 (1977), 1–16 | DOI | MR | Zbl

[56] Henriksen M., “Two classes of rings generated by their units”, J. Algebra, 31:1 (1974), 182–193 | DOI | MR | Zbl

[57] Hill P., “Endomorphism rings generated by units”, Trans. Am. Math. Soc., 141 (1969), 99–105 | DOI | MR | Zbl

[58] Khurana D., Srivastava A. K., “Right self-injective rings in which every element is a sum of two units”, J. Algebra Appl., 6:2 (2007), 281–286 | DOI | MR | Zbl

[59] Krylov P. A., Mikhalev A. V., Tuganbaev A. A., Endomorphism Rings of Abelian Groups, Springer-Verlag, Dordrecht–Boston–London, 2003 | MR

[60] Krylov P. A., Tuganbaev A. A., Modules over Discrete Valuation Domains, Walter de Gruyter, Berlin, 2008 | MR | Zbl

[61] Krylov P. A., Tuganbaev A. A., “Idempotent functors and localizations in the categories of modules and abelian groups”, J. Math. Sci., 183:3 (2012), 323-382 | DOI | MR | Zbl

[62] Krylov P. A., Tuganbaev A. A., Formal Matrices, Springer, Cham, 2017 | MR | Zbl

[63] Li Y., “Strongly clean matrix rings over local rings”, J. Algebra, 312:1 (2007), 397–404 | DOI | MR | Zbl

[64] Meehan C., “Sums of automorphisms of free abelian groups and modules”, Math. Proc. Royal Irish Acad., 104 (2004), 59–66 | DOI | MR

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

[66] Nicholson W. K., “Strongly clean rings and Fitting's lemma”, Commun. Algebra, 27:8 (1999), 3583–3592 | DOI | MR | Zbl

[67] Praeger C. E., Schultz P., “The Loewy length of the Jacobson radical of a bounded endomorphism ring”, Abelian Groups and Noncommutative Rings, A Collection of Papers in Memory of R. B. Warfield, Am. Math. Soc., Providence, Rhode Island, 1992, 349–360 | MR | Zbl

[68] Rangaswamy K. M., “Abelian groups with endomorphic images of special types”, J. Algebra, 6:3 (1967), 271–280 | DOI | MR | Zbl

[69] Raphael R. M., “Rings which are generated by their units”, J. Algebra, 28:1 (1974), 199–205 | DOI | MR | Zbl

[70] Richman F., Walker E. A., “Primary abelian groups as modules over their endomorphism rings”, Math. Z., 89:1 (1965), 77–81 | DOI | MR | Zbl

[71] Schultz P., “The endomorphism ring of the additive group of a ring”, J. Austr. Math. Soc., 15 (1973), 60–69 | DOI | MR | Zbl

[72] Skornyakov L. A., Complemented Modular Lattices and Regular Rings, Oliver Boyd, Edinburgh, 1963 | MR

[73] Srivastava A. K., “A survey of rings generated by units”, Ann. Fac. Sci. Toulouse Math., 6:19 (2010), 203–213 | DOI | MR | Zbl

[74] Tsarev A. V., “The pseudo-rational rank of quotient divisible group”, J. Math. Sci., 144:2 (2007), 4013–4022 | DOI | MR

[75] Tsarev A. V., “Modules over the ring of pseudo-rational numbers and quotient divisible groups”, St. Petersburg Math. J., 18 (2007), 657–669 | DOI | MR | Zbl

[76] Vámos P., “$2$-Good rings”, Quart. J. Math., 56:3 (2005), 417–430 | DOI | MR

[77] Wickless W., “A functor from mixed groups to torsion-free groups”, Contemp. Math., 171, 1994, 407–419 | DOI | MR

[78] Wickless W., “The Baer–Kaplansky theorem for direct sums of self-small mixed groups”, Abelian Groups and Modules, Birkhäuser, Basel, 1999, 101–106 | DOI | MR | Zbl

[79] Wickless W., “Direct sums of quotient divisible groups”, Commun. Algebra, 31:1 (2003), 79–96 | DOI | MR | Zbl

[80] Wolfson K. G., “An ideal-theoretic characterization of the ring of all linear transformations”, Am. J. Math., 75:2 (1953), 358–386 | DOI | MR | Zbl

[81] Yang X., Zhou Y., “Strong cleanness of the $2\times 2$ matrix ring over a general local ring”, J. Algebra, 320:6 (2008), 2280–2290 | DOI | MR | Zbl

[82] Zelinsky D., “Every linear transformation is a sum of nonsingular ones”, Proc. Am. Math. Soc., 5:4 (1954), 627–630 | DOI | MR | Zbl