On the socles of characteristically inert subgroups of Abelian p-groups
-
Andrey R. Chekhlov
and
Brendan Goldsmith
Abstract
We define the notion of a characteristically inert socle-regular Abelian p-group and explore such groups by focussing on their socles, thereby relating them to previously studied notions of socle-regularity. We show that large classes of p-groups, including all divisible, totally projective and torsion-complete p-groups, share this property when the prime p is odd. The present work generalizes notions of full inertia intensively studied recently by several authors and is a development of a recent work of the authors published in Mediterranean J. Math. (2021).
Funding statement: The work of the second author was supported in part by the Bulgarian National Science Fund under Grant KP-06 No. 32/1 of December 7, 2019.
References
[1] A. R. Chekhlov, Fully inert subgroups of completely decomposable finite rank groups and their commensurability, Vestn. Tomsk. Gos. Univ. Math. & Mech. 41 (2016), 42–50. Search in Google Scholar
[2] A. R. Chekhlov, On fully inert subgroups of completely decomposable groups, Math. Notes 101 (2017), no. 2, 365–373. 10.1134/S0001434617010394Search in Google Scholar
[3] A. R. Chekhlov, P. V. Danchev and B. Goldsmith, On the socles of fully inert subgroups of Abelian p-groups, Mediterranean J. Math. 18 (2021), no. 3. 10.1007/s00009-021-01747-zSearch in Google Scholar
[4] A. L. S. Corner, On endomorphism rings of primary abelian groups. II, Quart. J. Math. Oxford Ser. (2) 27 (1976), 5–13. 10.1093/qmath/27.1.5Search in Google Scholar
[5] A. L. S. Corner, The independence of Kaplansky’s notions of transitivity and full transitivity, Quart. J. Math. Oxford Ser. (2) 27 (1976), 15–20. 10.1093/qmath/27.1.15Search in Google Scholar
[6] P. V. Danchev and B. Goldsmith, On the socles of fully invariant subgroups of abelian p-groups, Arch. Math. (Basel) 92 (2009), no. 3, 191–199. 10.1007/s00013-009-3021-9Search in Google Scholar
[7] P. V. Danchev and B. Goldsmith, On the socles of characteristic subgroups of abelian p-groups, J. Algebra 323 (2010), no. 10, 3020–3028. 10.1016/j.jalgebra.2009.12.005Search in Google Scholar
[8] S. Files and B. Goldsmith, Transitive and fully transitive groups, Proc. Amer. Math. Soc. 126 (1998), no. 6, 1605–1610. 10.1090/S0002-9939-98-04330-5Search in Google Scholar
[9] L. Fuchs, Infinite Abelian Groups. Vol. I, Academic Press, New York, 1970. Search in Google Scholar
[10] L. Fuchs, Infinite Abelian Groups. Vol. II, Academic Press, New York, 1973. Search in Google Scholar
[11] L. Fuchs, Abelian Groups, Springer, Cham, 2015. 10.1007/978-3-319-19422-6Search in Google Scholar
[12] B. Goldsmith, S. Pabst and A. Scott, Unit sum numbers of rings and modules, Quart. J. Math. Oxford Ser. (2) 49 (1998), 331–344. 10.1093/qmathj/49.3.331Search in Google Scholar
[13] B. Goldsmith and L. Salce, Fully inert subgroups of torsion-complete p-groups, J. Algebra 555 (2020), 406–424. 10.1016/j.jalgebra.2020.02.038Search in Google Scholar
[14] B. Goldsmith and L. Salce, Abelian p-groups with minimal full inertia, Periodica Math. Hungar., to appear. 10.1007/s10998-021-00414-wSearch in Google Scholar
[15] B. Goldsmith, L. Salce and P. Zanardo, Fully inert subgroups of Abelian p-groups, J. Algebra 419 (2014), 332–349. 10.1016/j.jalgebra.2014.07.021Search in Google Scholar
[16] P. Hill, Endomorphism rings generated by units, Trans. Amer. Math. Soc. 141 (1969), 99–105. 10.1090/S0002-9947-1969-0242944-7Search in Google Scholar
[17] I. Kaplansky, Infinite Abelian Groups, University of Michigan, Ann Arbor, 1954 and 1969. Search in Google Scholar
[18] P. Keef, Countably totally projective abelian p-groups have minimal full inertia, J. Commut. Algebra, to appear. Search in Google Scholar
© 2021 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Higher depth mock theta functions and q-hypergeometric series
- Topological and algebraic properties of universal groups for right-angled buildings
- On the socles of characteristically inert subgroups of Abelian p-groups
- Priestley duality for MV-algebras and beyond
- The cardinality of μM,D‐orthogonal exponentials for the planar four digits
- Associated prime ideals of equivariant coinvariant algebras, Steenrod operations, and Krull’s Going Down Theorem
- Ordered fields dense in their real closure and definable convex valuations
- Third Hankel determinants for two classes of analytic functions with real coefficients
- A common range problem for model spaces
- Generalized Ricci flow on nilpotent Lie groups
- Endpoint estimates for a trilinear pseudo-differential operator with flag symbols
- The role of the algebraic structure in Wold-type decomposition
- Incidences between Euclidean spaces over finite fields
- Cancellation in algebraic twisted sums on GL_m
Articles in the same Issue
- Frontmatter
- Higher depth mock theta functions and q-hypergeometric series
- Topological and algebraic properties of universal groups for right-angled buildings
- On the socles of characteristically inert subgroups of Abelian p-groups
- Priestley duality for MV-algebras and beyond
- The cardinality of μM,D‐orthogonal exponentials for the planar four digits
- Associated prime ideals of equivariant coinvariant algebras, Steenrod operations, and Krull’s Going Down Theorem
- Ordered fields dense in their real closure and definable convex valuations
- Third Hankel determinants for two classes of analytic functions with real coefficients
- A common range problem for model spaces
- Generalized Ricci flow on nilpotent Lie groups
- Endpoint estimates for a trilinear pseudo-differential operator with flag symbols
- The role of the algebraic structure in Wold-type decomposition
- Incidences between Euclidean spaces over finite fields
- Cancellation in algebraic twisted sums on GL_m
