E-groups and E-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. 111-132.

Résumé

An associative ring $R$ is called an $E$-ring if the canonical homomorphism $R\cong \textsf{E}(R^+)$ is an isomorphism. Additive groups of $E$-rings are called $E$-groups. In other words, an Abelian group $A$ is an $E$-group if and only if $A\cong \operatorname{End} A$ and the endomorphism ring $\textsf{E}(A)$ is commutative. In this paper, we give a survey of the main results on $E$-groups and $E$-rings and also consider some of their generalizations: $\mathcal{E}$-closed groups, $T$-rings, $A$-rings, the groups admitting only commutative multiplications, etc.

Détail

Keywords: Abelian group, $\mathcal{E}$-closed group, $E$-ring, $T$-ring, $A$-ring, endomorphism ring.
Mots-clés : $E$-group, quotient divisible group

Bibliographie

[1] Grishin A. V., Timoshenko E. A., Tsarev A. V., “Posledovatelnosti grupp endomorfizmov abelevykh grupp”, Mat. zametki (to appear)

[2] Krylov P. A., Tuganbaev A. A., “Idempotentnye funktory i lokalizatsii v kategoriyakh modulei i abelevykh grupp”, Fundam. prikl. mat., 16:7 (2010), 75–159

[3] Grishin A. V., Tsarev A. V., “$\mathcal E$-Zamknutye gruppy i moduli”, Fundam. prikl. mat., 17:2 (2012), 97–106 | Zbl

[4] Davydova O. I., “Faktorno delimye abelevy gruppy ranga 1”, Fundam. prikl. mat., 13:3 (2007), 25–33 | MR

[5] Karpov O. A., Abelevy gruppy, dopuskayuschie tolko kommutativnye umnozheniya, v pechati

[6] Prikhodovskii M. A., Izomorfizmy tenzornykh proizvedenii modulei i $T$-moduli, Diss. na soiskanie uch. stepeni kand. fiz.-mat. nauk, Tomsk, 2002

[7] E. A. Timoshenko, “$T$-Radikaly i $E$-radikaly v kategorii modulei”, Sib. mat. zh., 45:1 (2004), 201–210 | MR | Zbl

[8] Fomin A. A., “Abelevy gruppy so svobodnymi podgruppami beskonechnogo indeksa i ikh koltsa endomorfizmov”, Mat. zametki, 36:2 (1984), 179–187 | MR | Zbl

[9] Fomin A. A., “Servantno svobodnye gruppy”, Abelevy gruppy i moduli, v. 6, Tomsk, 1986, 145–164

[10] Friger M. D., “O zhestkikh koltsakh bez krucheniya”, Sib. mat. zh., 27:3 (1986), 217–219 | MR | Zbl

[11] Tsarev A. V., “$T$-Koltsa i faktorno delimye gruppy ranga $1$”, Vestn. Tomsk. gos. un-ta. Mat. mekh., 2013, no. 4, 50–53

[12] Arnold D. M., Finite rank torsion free abelian groups and rings, Lect. Notes Math., 931, Springer, New York, 1982 | DOI | MR | Zbl

[13] Beaumont R. A., Pierce R. S., “Subrings of algebraic number fields”, Acta Sci. Math. Szeged, 22 (1961), 202–216 | MR | Zbl

[14] Beaumont R. A., Pierce R. S., “Torsion free rings”, Ill. J. Math., 5:1 (1961), 61–98 | MR | Zbl

[15] Beaumont R. A., Pierce R. S., “Isomorphic direct summands of abelian groups”, Math. Ann., 153:1 (1964), 21–37 | DOI | MR

[16] Blagoveshchenskaya E., Ivanov G., Schultz P., “The Baer–Kaplansky theorem for almost completely decomposable groups”, Contemp. Math., 273, 2001, 85–93 | DOI | MR | Zbl

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

[18] Dugas M., “$AA$-Rings”, Commun. Algebra, 32:10 (2004), 3853–3860 | DOI | MR | Zbl

[19] Dugas M., Feigelstock S., “$A$-Rings”, Colloq. Math., 96:2 (2003), 277–292 | DOI | MR | Zbl

[20] Dugas M., Vinsonhaler C., “Two-side $E$-rings”, J. Pure Appl. Algebra, 185 (2003), 87–102 | DOI | MR | Zbl

[21] Feigelstock S., “Full subrings of $E$-rings”, Bull. Austr. Math. Soc., 54 (1996), 275–280 | DOI | MR | Zbl

[22] Feigelstock S., “Additive groups of commutative rings”, Quaest. Math., 23:2 (2000), 241–245 | DOI | MR | Zbl

[23] Feigelstock S., Hausen J., Raphael R., “Groups which map onto their endomorphism rings”, Proc. Dublin Conf. (1998), Basel, 1999, 231–241 | MR

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

[25] Fuchs L., Abelian Groups, Pergamon Press, New York–Oxford–London–Paris, 1960 | MR | Zbl

[26] Fuchs L., Infinite Abelian Groups, v. I, Academic Press, New York, 1970 | MR | Zbl

[27] Fuchs L., Infinite Abelian Groups, v. II, Academic Press, New York, 1973 | MR | Zbl

[28] Göbel R., Shelah S., Strüngmann L., Generalized $E$-Rings, 2003, arXiv: math/0404271 [math.LO]

[29] Grishin A. V., “Strongly indecomposable localizations of the ring of algebraic integers”, Commun. Algebra, 43:9 (2015), 3816–3822 | DOI | MR | Zbl

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

[31] Mader A., Vinsonhaler C., “Torsion-free $E$-modules”, J. Algebra, 115:2 (1988), 401–411 | DOI | MR | Zbl

[32] Pierce R. S., “$E$-Modules”, Abelian Group Theory, Contemp. Math., 87, Am. Math. Soc., Providence, Rhode Island, 1989, 221–240 | DOI | MR | Zbl

[33] Schultz P., “Periodic homomorphism sequences of abelian groups”, Arch. Math., 21 (1970), 132–135 | DOI | MR | Zbl

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

[35] Vinsonhaler C., “$E$-Rings and related structures”, Non-Noetherian Commutative Ring Theory, Math. Appl., 520, Kluwer, Dordrecht, 2002, 387–402 | MR

[36] Wilson G. V., “Additive groups of $T$-rings”, Proc. Am. Math. Soc., 99:2 (1987), 219–220 | MR | Zbl