Singly Generated Socles and Radicals

Fay, Temple H.; Oxford, Edwin P.; Walls, Gary L.

doi:10.1007/978-3-662-21560-9_48

Cited by 7 publications

(2 citation statements)

References 3 publications

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“…We have found no example of this situation in the literature, so we describe one in the sequel: Proof. It is immediate from the previous discussion and Example 6.2 of [20] (see also [14]). ✷…”

Section: Comparing the Classesmentioning

confidence: 80%

Generators and closed classes of groups

Flores¹,

Rodríguez²

2017

Preprint

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We show that in the category of groups, every singly-generated class which is closed under isomorphisms, direct limits and extensions is also singly-generated under isomorphisms and direct limits, and in particular is co-reflective. In this way, we extend to the group-theoretic framework the topological analogue proved by Chachólski, Parent and Stanley in 2004. We also establish several new relations between singly-generated closed classes. IntroductionIn a category of R-modules over a certain ring R, the study of the closure of different classes under different operations has been a hot research topic for decades (see [16] for a thorough approach). The operations include extensions, subgroups, quotients, etc. In the case of R = Z, we are dealing, of course, with operations in the category of abelian groups.There is another different but related problem. Given a collection D of objects in a class C, the smallest class that contains D and is closed under certain operations is called the class generated by D under these operations. When this class is equal to C, it is said that C is generated by D under the operations, and the elements of D are called generators of C. In this situation, if D consists of only one object, the class C is said to be

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Section: Comparing the Classesmentioning

confidence: 80%

Generators and closed classes of groups

Flores¹,

Rodríguez²

2017

Preprint

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“…It follows that pW = Rnz(,,) = ReZ(pn). In [4], it was shown that for any ordinal a > w, the radical pa is not cogenerated by a single group, and in [7] it was shown that the radical p" is not cogenerated by a single group. In fact, we will observe that if R is a radical having p" 5 R < for X > w, then R # Rx for any group X.…”

Section: Preliminariesmentioning

confidence: 98%

Radicals of abelian groups

Cutler¹,

Fay²

1990

Communications in Algebra

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The descripti4 ,ns of torsion 1 lreories a.nd of torsion vnlurd itlernpotent preradicals of a.bc.lian groups are well known. T l~c descri~~tion of all radicals of abclian groups is far from known and it is tho purpose of this paper to tlemonstrat,e Ilia(. the description of radicals whosc vslacs are torsion groups may he as complicatetl as tho tlescri~~tion of all torsion groups t.l~emsrlvcs. For each nou-rtegatiie integcr n , we coilsidcr singly cog'rieratctl radir;ils which fit betwcer~ the radicals pW+'"+' and pW+7L, showing t,h;t there is a proper class ol' sucl~ radicals forming ;L chain therein. We also exar~li~re oll~er intervals in the lattice of radicals sliowing either gaps or thc existence of singly cogencratrd ratliritls in these intervals.

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Cellular approximations using Moore spaces

Rodríguez

Scherer

2001

Cohomological Methods in Homotopy Theory

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Singly Generated Socles and Radicals

Cited by 7 publications

References 3 publications

Generators and closed classes of groups

Generators and closed classes of groups

Radicals of abelian groups

Cellular approximations using Moore spaces

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