| Title:
|
Self-small products of abelian groups (English) |
| Author:
|
Dvořák, Josef |
| Author:
|
Žemlička, Jan |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
63 |
| Issue:
|
2 |
| Year:
|
2022 |
| Pages:
|
145-157 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $A$ and $B$ be two abelian groups. The group $A$ is called $B$-small if the covariant functor ${\rm Hom}(A,-)$ commutes with all direct sums $B^{(\kappa)}$ and $A$ is self-small provided it is $A$-small. The paper characterizes self-small products applying developed closure properties of the classes of relatively small groups. As a consequence, self-small products of finitely generated abelian groups are described. (English) |
| Keyword:
|
self-small abelian group |
| Keyword:
|
slender group |
| MSC:
|
20K20 |
| MSC:
|
20K21 |
| MSC:
|
20K40 |
| idZBL:
|
Zbl 07613027 |
| idMR:
|
MR4506129 |
| DOI:
|
10.14712/1213-7243.2022.020 |
| . |
| Date available:
|
2022-11-02T09:11:50Z |
| Last updated:
|
2024-07-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/151082 |
| . |
| Reference:
|
[1] Albrecht U., Breaz S.: A note on self-small modules over RM-domains.J. Algebra Appl. 13 (2014), no. 1, 1350073, 8 pages. MR 3096854, 10.1142/S0219498813500734 |
| Reference:
|
[2] Albrecht U., Breaz S., Schultz P.: Functorial properties of Hom and Ext.in: Groups and Model Theory, Contemp. Math., 576, Amer. Math. Soc., Providence, 2012, pages 1–15. MR 2962871 |
| Reference:
|
[3] Albrecht U., Breaz S., Wickless W.: Purity and self-small groups.Comm. Algebra 35 (2007), no. 11, 3789–3807. MR 2362684, 10.1080/00927870701509545 |
| Reference:
|
[4] Albrecht U., Breaz S., Wickless W.: Self-small abelian groups.Bull. Aust. Math. Soc. 80 (2009), no. 2, 205–216. MR 2540354, 10.1017/S0004972709000185 |
| Reference:
|
[5] Arnold D. M., Murley C. E.: Abelian groups, $A$, such that $ Hom(A, - - -)$ preserves direct sums of copies of $A$.Pacific J. Math. 56 (1975), no. 1, 7–20. MR 0376901, 10.2140/pjm.1975.56.7 |
| Reference:
|
[6] Bass H.: Algebraic $K$-theory.Mathematics Lecture Note Series, W. A. Benjamin, New York, 1968. MR 0249491 |
| Reference:
|
[7] Breaz S.: Self-small abelian groups as modules over their endomorphism rings.Comm. Algebra 31 (2003), no. 10, 4911–4924. MR 1998035, 10.1081/AGB-120023139 |
| Reference:
|
[8] Breaz S.: A mixed version for a Fuchs’ lemma.Rend. Semin. Mat. Univ. Padova 144 (2020), 61–71. MR 4186446, 10.4171/RSMUP/56 |
| Reference:
|
[9] Breaz S., Schultz P.: Dualities for self-small groups.Proc. Amer. Math. Soc. 140 (2012), no. 1, 69–82. MR 2833518, 10.1090/S0002-9939-2011-10919-5 |
| Reference:
|
[10] Breaz S., Žemlička J.: When every self-small module is finitely generated.J. Algebra 315 (2007), no. 2, 885–893. MR 2351899, 10.1016/j.jalgebra.2007.01.037 |
| Reference:
|
[11] Colpi R., Menini C.: On the structure of $\star$-modules.J. Algebra 158 (1993), no. 2, 400–419. MR 1226797, 10.1006/jabr.1993.1138 |
| Reference:
|
[12] Dvořák J.: On products of self-small abelian groups.Stud. Univ. Babeş–Bolyai Math. 60 (2015), no. 1, 13–17. MR 3335780 |
| Reference:
|
[13] Dvořák J., Žemlička J.: Autocompact objects of Ab5 categories.Theory Appl. Categ. 37 (2021), Paper No. 30, 979–995. MR 4326106 |
| Reference:
|
[14] Fuchs L.: Infinite Abelian Groups. Vol. I.Pure and Applied Mathematics, 36, Academic Press, New York, 1970. Zbl 0338.20063, MR 0255673 |
| Reference:
|
[15] Fuchs L.: Infinite Abelian Groups. Vol. II.Pure and Applied Mathematics, 36-II, Academic Press, New York, 1973. MR 0349869 |
| Reference:
|
[16] Gómez Pardo J. L., Militaru G., Năstăsescu C.: When is $\mathrm{HOM}_R(M,-)$ equal to $\mathrm{Hom}_R(M,-)$ in the category $R-gr$?.Comm. Algebra 22 (1994), no. 8, 3171–3181. MR 1272380 |
| Reference:
|
[17] Kálnai P., Žemlička J.: Compactness in abelian categories.J. Algebra 534 (2019), 273–288. MR 3979075, 10.1016/j.jalgebra.2019.05.037 |
| Reference:
|
[18] Modoi G. C.: Constructing large self-small modules.Stud. Univ. Babeş–Bolyai Math. 64 (2019), no. 1, 3–10. MR 3928597, 10.24193/subbmath.2019.1.01 |
| Reference:
|
[19] Rentschler R.: Sur les modules $M$ tels que $ Hom(M,-)$ commute avec les sommes directes.C. R. Acad. Sci. Paris Sér. A-B 268 (1969), A930–A933 (French). MR 0241466 |
| Reference:
|
[20] Žemlička J.: When products of self-small modules are self-small.Comm. Algebra 36 (2008), no. 7, 2570–2576. MR 2422503, 10.1080/00927870802070207 |
| . |
