The endomorphism ring of the additive group of a ring

Schultz, Phill

doi:10.1017/s1446788700012763

Cited by 82 publications

(40 citation statements)

References 2 publications

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“…As a preparation for the findings in Section 6, we shall now discuss a more general notion, namely rigid algebras. Most of the following results generalize basic properties of rigid rings that can be found in [26] or [9]. Some observations are new, notably Theorem 4.3.…”

Section: Rigid Rings and Algebrassupporting

confidence: 53%

“…We summarize our conclusions in the following theorem. Observe that, if R is a rigid ring and the unit map Z → R is not injective, then R ∼ = Z/t for some integer t. Indeed, if the identity element of R has finite order, then tR = 0 for some integer t and, for a rigid ring, this implies that R is cyclic; this fact was already noted in [26]. Therefore, if L f K(Z, n) K(R, n), then either On the other hand, all finite-dimensional CW-complexes are f -local by Miller's main theorem in [19], yet their homotopy groups need not be Z[1/p]-modules.…”

Section: Effect On the Higher Homotopy Groupsmentioning

confidence: 77%

“…We prove that L f K(Z/p r , n) = K(Z/p i p (f ) , n) if n = n p (f ) and r > i p (f ) R is any commutative ring, we say that an R-algebra A with 1 is rigid if evaluation at 1 yields an isomorphism of R-algebras Hom R (A, A) ∼ = A. Some basic properties of rigid R-algebras are described in Section 4, where we extend earlier results of Bowshell and Schultz [9], [26]. Rigid rings have previously been investigated under the name of E-rings.…”

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confidence: 78%

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Localizations of abelian Eilenberg–Mac Lane spaces of finite type

Casacuberta

Rodríguez

Tai

2016

Algebr. Geom. Topol.

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Abstract. Using recent techniques of unstable localization, we extend earlier results on homological localizations of Eilenberg-Mac Lane spaces, and show that several deep properties of such localizations can be explained by the preservation of certain algebraic structures under the effect of idempotent functors.We study localizations L f K(G, n) of Eilenberg-Mac Lane spaces with respect to any map f , where n ≥ 1 and G is abelian. We find that, if G is finitely generated, then the result is a K(A, n), where A can be computed using cohomological data derived from f . If G = Z, then A is a commutative ring which is isomorphic to the ring End(A) of its own additive endomorphisms; such rings, which we call rigid, form a proper class which contains the set of solid rings. From this fact it follows that there is a proper class of distinct homotopical localizations of the circle S 1 . Among other applications of our results, we show that, if X is a product of abelian Eilenberg-Mac Lane spaces and f is any map, then the homotopy groups π m (L f X) become modules over the ring π 1 (L f S 1 ). IntroductionWe refer the reader to [5] and [15] for the essentials of homotopical localization.Given any map f : W → V between CW-complexes, for each space X there is a map η: X → L f X where L f X is universal in the homotopy category with the property that the map of function spacesinduced by f is a weak homotopy equivalence. The space L f X is called the f -localization of X.As explained in [15, 4.B], if f is any map, G is any abelian group, and n ≥ 1, then the f -localization of an Eilenberg-Mac Lane space K(G, n) has at most two nontrivial homotopy groups, and it is in fact a product K(A, n) × K(B, n + 1). In the present article, for any map f : W → V , we firstly compute the homotopy groups A and B of the f -localization L f K(G, n) in terms of f when G is finitely generated; in fact, in this case it happens that the group B is necessarily zero.Since localization commutes with finite direct products, it suffices to consider the case when G is cyclic. If G = Z/p r , then we prove that either A = 0 or A = Z/p j for some j ≤ r. This was first observed by Chachólski in the case when V is contractible; under this assumption on V , one finds that necessarily j = r.Contrary to this fact, if no restriction is imposed on the spaces V and W , then each j ≤ r can occur, and the value of j is determined by two (possibly infinite)

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Section: Rigid Rings and Algebrassupporting

confidence: 53%

Section: Effect On the Higher Homotopy Groupsmentioning

confidence: 77%

mentioning

confidence: 78%

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Localizations of abelian Eilenberg–Mac Lane spaces of finite type

Casacuberta

Rodríguez

Tai

2016

Algebr. Geom. Topol.

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“…• A consequence of (a) is that if A is a reduced mixed group of torsion-free rank one, then various conditions on either the endormorphism ring of A, or the rings supported by A force A to be in /Z Examples abound in the literature, see for example Fuchs [2], Fuchs and Rangaswamy [4], Rangaswamy [9], Schultz [11], and Szele and Szendrei [13]. …”

Section: A P Is the ^-Primary Component Of T(a) The Torsion Subgroupmentioning

confidence: 99%

Rings on certain mixed abelian groups

Jackett¹

1982

Pacific J. Math.

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“…A (discrete) valuation ¿s-ring is a (discrete) valuation ring which is also an £-ring. £-rings were introduced by P. Schultz [10] as the rings whose additive endomorphisms are given by left multiplication with elements. Throughout, E(G) is the endomorphism ring of G, the center of a ring R is denoted by Cent R, and Mat",(/?)…”

mentioning

confidence: 99%