Theorem Of Kaplansky Research Articles

On the Baer–Kaplansky theorem for injective modules

On the Baer–Kaplansky theorem for injective modules

Influence of the Baer–Kaplansky Theorem on the Development of the Theory of Groups, Rings, and Modules

The review presents an analysis of results of the Abelian group theory, as well as rings and modules, which concern the definability of algebraic structures by their endomorphism rings and related structures. In the systematization of the results, the greatest attention is paid to torsion-free Abelian groups, which are of particular interest due to the presence of non-isomorphic direct decompositions in this class. This significantly expands the understanding of general, including modern, trends of the development of algebra in the context related to the Baer–Kaplansky theorem. The reflection of the properties of algebraic objects of a certain class in their endomorphism rings is a natural structural connection, the study of which is a separate investigation direction. A striking introduction to this topic was the Baer–Kaplansky theorem for torsion Abelian groups, which dates back to the middle of the last century and states that any isomorphism of endomorphism rings of two groups from this class is inevitably induced by some isomorphism of the groups themselves. Of course, it follows that if two torsion Abelian groups have isomorphic endomorphism rings, then they are isomorphic. This profound result inspired mathematicians to obtain results in the same form concerning other classes of objects. But even in the theory of Abelian groups itself, other classes were discovered for which the analogue of the Baer–Kaplansky theorem is valid. Despite the fundamental difference between the definitions of completely decomposable Abelian groups, which are direct sums of rank-one torsion-free groups, and torsion Abelian groups considered, which are direct sums of finite order cyclic groups, there is one very important common characteristic of these classes: their decompositions into indecomposable summands are determined uniquely up to isomorphism. This property is not possessed by torsion-free Abelian groups in general, whose definability by their endomorphism rings is in the focus of our attention.

ON THE CLASS OF $n$-NORMAL OPERATORS AND MOORE-PENROSE INVERSE

Let $ T \in B(H)$ be a bounded linear operator on a complex Hilbert space $H$. For $ n\in \mathbb{N } $, an operator $ T\in B(H)$ is said to be n-normal if $ T^{n}T^{*}=T^{*}T^{n} $. In this paper we investigate a necessary and sufficient condition for the n-normality of $ ST $ and $ TS $, where $ S,T \in B(H). $ As a consequence, we generalize Kaplansky theorem for normal operators to n-normal operators. Also, In this paper, we provide new characterizations of n-normal operators by certain conditions involving powers of Moore-Penrose inverse.

Around the Baer–Kaplansky Theorem

Around the Baer–Kaplansky Theorem

The Baer–Kaplansky Theorem for all abelian groups and modules

It is shown that the Baer–Kaplansky Theorem can be extended to all abelian groups provided that the rings of endomorphisms of groups are replaced by trusses of endomorphisms of corresponding heaps. That is, every abelian group is determined up to isomorphism by its endomorphism truss and every isomorphism between two endomorphism trusses associated to some abelian groups [Formula: see text] and [Formula: see text] is induced by an isomorphism between [Formula: see text] and [Formula: see text] and an element from [Formula: see text]. This correspondence is then extended to all modules over a ring by considering heaps of modules. It is proved that the truss of endomorphisms of a heap associated to a module [Formula: see text] determines [Formula: see text] as a module over its endomorphism ring.

One-sided central Drazin inverses

ABSTRACT We study the one-sided version of central Drazin inverses. An element a in a ring R is said to be left central Drazin invertible if there is such that , for some . The right central Drazin invertible elements can be defined similarly. It is shown that an element is central Drazin invertible if and only if a is both left and right central Drazin invertible. Some well-known results on Drazin inverses and left invertible elements including the famous Kaplansky theorem in [Jacobson N. Some remarks on one-sided inverses. Proc Amer Math Soc. 1950;1:352–355] are generalized. As applications, we give a new characterization of Dedekind-finite rings from the point of view of one-sided central Drazin invertible elements. The Cline's formula on Drazin invertible elements is also generalized.

A new version of a theorem of Kaplansky

A well-known theorem of Kaplansky states that any projective module is a direct sum of countably generated modules. In this paper, we prove the w-version of this theorem, where w is a hereditary torsion theory for modules over a commutative ring.Communicated by Silvana Bazzoni

On linear operators for which TTD is normal

A Drazin invertible operator T ? B(H) is called skew D-quasi-normal operator if T* and TTD commute or equivalently TTD is normal. In this paper, firstly we give a list of conditions on an operator T; each of which is equivalent to T being skew D-quasi-normal. Furthermore, we obtain the matrix representation of these operators. We also develop some basic properties of such operators. Secondly we extend the Kaplansky theorem and the Fuglede-Putnam commutativity theorem for normal operators to skew D-quasi-normal matrices.

Connections between representation-finite and Köthe rings

A ring R is called left k-cyclic if every left R-module is a direct sum of indecomposable modules which are homomorphic image of RkR. In this paper, we give a characterization of left k-cyclic rings. As a consequence, we give a characterization of left Köthe rings, which is a generalization of Köthe–Cohen–Kaplansky theorem. We also characterize rings which are Morita equivalent to a basic left k-cyclic ring. As a corollary, we show that R is Morita equivalent to a basic left Köthe ring if and only if R is an artinian left multiplicity-free top ring.

Order isomophisms between Riesz spaces

The first aim of this paper is to give a description of the (not necessarily linear) order isomorphisms C(X)rightarrow C(Y) where X, Y are compact Hausdorff spaces. For a simple case, suppose X is metrizable and T is such an order isomorphism. By a theorem of Kaplansky, T induces a homeomorphism tau :Xrightarrow Y. We prove the existence of a homeomorphism Xtimes mathbb {R}rightarrow Ytimes mathbb {R} that maps the graph of any fin C(X) onto the graph of Tf. For nonmetrizable spaces the result is similar, although slightly more complicated. Secondly, we let X and Y be compact and extremally disconnected. The theory of the first part extends directly to order isomorphisms C^{infty }(X)rightarrow C^{infty }(Y). (Here C^{infty }(X) is the space of all continuous functions Xrightarrow [-infty ,infty ] that are finite on a dense set.) The third part of the paper considers order isomorphisms T between arbitrary Archimedean Riesz spaces E and F. We prove that such a T extends uniquely to an order isomorphism between their universal completions. (In the absence of linearity this is not obvious.) It follows, that there exist an extremally disconnected compact Hausdorff space X, Riesz isomorphisms hat{} of E and F onto order dense Riesz subspaces of C^{infty }(X) and an order isomorphism S:C^{infty }(X)rightarrow C^{infty }(X) such that hat{Tf}=Shat{f} (fin E).

About G-rings

In this paper, we are concerned with G-rings. We generalize the Kaplansky's theorem to rings with zero-divisors. Also, we assert that if $R \subseteq T$ is a ring extension such that $mT\subseteq R$ for some regular element $m$ of $T$, then $T$ is a G-ring if and only if so is $R$. Also, we examine the transfer of the G-ring property to trivial ring extensions. Finally, we conclude the paper with illustrative examples discussing the utility and limits of our results.

An extension of a theorem of Kaplansky

ABSTRACTA theorem of Kaplansky asserts that a semigroup of matrices with entries from a field whose members all have singleton spectra is triangularizable. Indeed, Kaplansky’s Theorem unifies well-known theorems of Kolchin and Levitzki on simultaneous triangularizability of semigroups of unipotent and nilpotent matrices, respectively. First, for a division ring D of characteristic zero whose center intersects its multiplicative commutator group in a finite group, we prove that the counterpart of Kolchin’s Theorem over D implies that of Kaplansky’s Theorem over D. Next, we note that this proof, when adjusted in the setting of fields, provides a new and simple proof of Kaplansky’s Theorem over fields of characteristic zero. We show that if Kaplansky’s Theorem holds over a division ring D, which is for instance the case over general fields, then a generalization of Kaplansky’s Theorem holds over D, and in particular over general fields.

A theorem of Kaplansky revisited

We present a simple proof of a theorem due to Kaplansky which unifies theorems of Kolchin and Levitzki on triangularizability of semigroups of matrices. We also give two different extensions of the theorem. As a consequence, we prove the counterpart of Kolchin's Theorem for finite groups of unipotent matrices over division rings.

FAMILIES OF ULTRAFILTERS, AND HOMOMORPHISMS ON INFINITE DIRECT PRODUCT ALGEBRAS

Abstract Criteria are obtained for a filter ${\cal F}$ of subsets of a set I to be an intersection of finitely many ultrafilters, respectively, finitely many κ-complete ultrafilters for a given uncountable cardinal $\kappa .$ From these, general results are deduced concerning homomorphisms on infinite direct product groups, which yield quick proofs of some results in the literature: the Łoś–Eda theorem (characterizing homomorphisms from a not-necessarily-countable direct product of modules to a slender module), and some results of Nahlus and the author on homomorphisms on infinite direct products of not-necessarily-associative k-algebras. The same tools allow other results of Nahlus and the author to be nontrivially strengthened, and yield an analog to one of their results, with nonabelian groups taking the place of k-algebras. We briefly examine the question of how the common technique used in applying the general results of this note to k-algebras on the one hand, and to nonabelian groups on the other, might be extended to more general varieties of algebras in the sense of universal algebra. In a final section, the Erdős–Kaplansky theorem on dimensions of vector spaces ${D^I}$ $(D$ a division ring) is extended to reduced products ${D^I}/{\cal F},$ and an application is noted.

Kaplansky Theorem for completely regular spaces

Let X , Y X, Y be realcompact spaces or completely regular spaces consisting of G δ G_\delta -points. Let ϕ \phi be a linear bijective map from C ( X ) C(X) (resp. C b ( X ) C^b(X) ) onto C ( Y ) C(Y) (resp. C b ( Y ) C^b(Y) ). We show that if ϕ \phi preserves nonvanishing functions, that is, \[ f ( x ) ≠ 0 , ∀ x ∈ X , ⟺ ϕ ( f ) ( y ) ≠ 0 , ∀ y ∈ Y , f(x)\neq 0,\forall \, x\in X, \quad \Longleftrightarrow \quad \phi (f)(y)\neq 0, \forall \, y\in Y, \] then ϕ \phi is a weighted composition operator \[ ϕ ( f ) = ϕ ( 1 ) ⋅ f ∘ τ , \phi (f)=\phi (1)\cdot f\circ \tau , \] arising from a homeomorphism τ \tau from Y Y onto X X . This result is applied also to other nice function spaces, e.g., uniformly or Lipschitz continuous functions on metric spaces.

Generalizations of Kaplansky's Theorem Involving Unbounded Linear Operators

Generalizations of Kaplansky's Theorem Involving Unbounded Linear Operators

A Lazard-Like Theorem for Quasi-Coherent Sheaves

We study filtrations of quasi–coherent sheaves. We prove a version of Kaplansky Theorem for quasi–coherent sheaves, by using Drinfeld’s notion of almost projective module and the Hill Lemma. We also show a Lazard-like theorem for flat quasi–coherent sheaves for quasi–compact and semi–separated schemes which satisfy the resolution property.

An analogue of Hilbert’s Syzygy Theorem for the algebra of one-sided inverses of a polynomial algebra

An analogue of Hilbert's Syzygy Theorem is proved for the algebra $\mS_n (A)$ of one-sided inverses of the polynomial algebra $A[x_1, ..., x_n]$ over an arbitrary ring $A$: $$ \lgldim (\mS_n(A))= \lgldim (A) +n.$$ The algebra $\mS_n(A)$ is noncommutative, neither left nor right Noetherian and not a domain. The proof is based on a generalization of the Theorem of Kaplansky (on the projective dimension) obtained in the paper. As a consequence it is proved that for a left or right Noetherian algebra $A$: $$ \wdim (\mS_n(A))= \wdim (A) +n.$$

A Kaplansky theorem for JB*-triples

A Kaplansky theorem for JB*-triples

A generalization of the Jaffard–Ohm–Kaplansky Theorem

The well-known Jaffard–Ohm–Kaplansky Theorem states that every abelian l-group can be realized as the group of divisibility of a commutative Bezout domain. To date there is no realization (except in certain circumstances) of an arbitrary, not necessarily abelian, l-group as the group of divisibility of an integral domain. We show that using filters on lattices we can construct a nice quantal frame whose “group of divisibility” is the given l-group. We then show that our construction when applied to an abelian l-group gives rise to the lattice of ideals of any Prufer domain assured by the Jaffard–Ohm–Kaplansky Theorem. Thus, we are assured of the appropriate generalization of the Jaffard–Ohm–Kaplansky Theorem.