Baer subrings of the ring of linear transformations

Wolfson, Kenneth G.

doi:10.1007/bf01211031

Cited by 5 publications

(5 citation statements)

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“…A ring R is called a Baer ring iff every annihilating right or left ideal of R is generated by an idempotent. In a recent paper [2], Wolfson proved the following result.…”

Section: Projectors Let J Be a Pdmentioning

confidence: 95%

Distinguished Rings of Linear Transformations

Johnson¹

1964

Transactions of the American Mathematical Society

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A ring R of linear transformations of a vector space M over a division ring D is called distinguished iff (1) the lattice J of all P-submodules of M is a distributive sublattice of the lattice L of all subspaces of M, and (2) the set of all linear transformations of M leaving J invariant is R. The study of such rings is motivated by a paper of Wolfson [2] in which J is a chain and by several recent papers of Behrens [4]. Behrens studies Artinian rings R with unity having faithful Rmodules M such that the lattice of P-submodules of M is distributive. Our primary interest is with distinguished rings R for which J is finite as well as distributive. Such a condition does not force the ring R to be Artinian. A basic tool in our study is a lattice theorem (1.1) stating that every element of J is a direct sum of elements of L associated with the irreducible elements of J. It is shown that every finite distributive sublattice of L containing 0 and M is the lattice of submodules of a distinguished ring R. If D has characteristic 0, then a finite sublattice of L must be distributive in order to be the lattice of submodules of a ring of linear transformations of M.

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“…A ring R is called a Baer ring iff every annihilating right or left ideal of R is generated by an idempotent. In a recent paper [2], Wolfson proved the following result.…”

Section: Projectors Let J Be a Pdmentioning

confidence: 95%

Distinguished Rings of Linear Transformations

Johnson¹

1964

Transactions of the American Mathematical Society

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“…In [278] Wolfson proves that if A and B are skew fields, then any ring isomorphism of the rings of triangular matrices T n {A) and T m (B) implies m = n and A and B isomorphic, provided m or n at least three (note that in [31] and [57] it was shown that any automorphism of the ring T n (A) fixing scalar matrices is inner). In [1] automorphisms of the ring of triangular endomorphisms of countable dimensional linear spaces were described.…”

Section: Isomorphisms and Automorphisms Of Triangular And Structural mentioning

confidence: 97%

Isomorphisms and anti-isomorphisms of endomorphism rings of modules

Mikhalev¹

1996

First International Tainan-Moscow Algebra Workshop

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Let A and B be rings and Ma and Ng a right A-module and a right 5-module respectively. Let a:End(M,4) -»• End(Ar fl ) be either an isomorphism or an anti-isomorphism of their endomorphism rings. The Wedderburn-Baer-Kaplansky problem is to characterize all such a. The problem has a long history. The aim of these lectures is to consider main results in this direction for modules Ma and Ng which are close to free ones (in the epicenter we put strong generators, i.e., modules with a cyclic free direct summand). The problems under consideration are connected with conditions for inducing of isomorphisms of endomorphism rings by semilinear transformations, by Morita equivalences or by full embeddings. We consider also multiplicative isomorphisms of endomorphism semigroups of modules. We touch some related topics. First of all, we describe recent results on isomorphisms of general linear groups over arbitrary associative rings (Schreier -van der Waerden problem), because of a nice reduction of this group problem to the ring problem of classification of isomorphisms and anti-isomorphisms of matrix rings. We consider Malcev type theorems on elementary equivalences of linear groups and rings of matrices. At the same time we deal with the characterization problem of endomorphism rings of modules from different classes (in particular, of matrix rings). Some comments are related to operator rings and semigroups of topological and ordered linear spaces and to Frobenius type results on A-linear mappings on matrix rings A n .

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“…Under some additional conditions, an arbitrary Baer factor of type I is of this form [29]. For other examples of Baer rings see [13,9,10,28].…”

Section: 2mentioning

confidence: 99%

On some classes of non-Archimedean operator algebras

Kochubei¹

2013

Advances in Ultrametric Analysis

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We study some classes of algebras of operators on non-Archimedean Banach spaces. In particular, we propose a non-Archimedean version of the crossed product construction.c 2012 American Mathematical Society 2.1. Spectral theorem. Let A be a bounded linear operator on a Banach space B over a complete non-Archimedean valued field K with a nontrivial valuation; | · | will denote the absolute value in K, O is the ring of integers in K. We denote by · both the norm in B and the operator norm A = supx =0 Ax x (see [21] for some subtleties regarding the operator norm in the non-Archimedean case). Below we assume that K is algebraically closed. We will denote by K the residue field of K.Denote by L A the commutative Banach algebra generated by A and the unit operator I. L A is the closure of the algebra K[A] of polynomials in A, with respect to the norm of operators; thus L A is a Banach subalgebra of the algebra L(B) of all bounded linear operators. Elements λ ∈ K are identified with the operators λI.The spectrum M(L A ) of the algebra L A is defined (see [3]) as the set of all bounded multiplicative seminorms | · | on L A . M(L A ) is provided with the weakest topology, with respect to which all functions | · | → |B|, B ∈ L A , are continuous. In this topology, it is a nonempty Hausdorff compact topological space. If the algebra L A is uniform, that is T 2 = T 2 for any T ∈ L A , and all the characters take their values in K, then [3] the space M(L A ) is totally disconnected, and L A is isomorphic to the algebra C(M(L A ), K) of continuous functions on M(L A ) with values in K. In this case the above isomorphism transforms the characteristic functions η A of nonempty open-closed subsets Λ ⊂ M(L A ) into idempotent operators E(Λ) ∈ L A , E(Λ) = 1. These operators form a finitely additive norm-bounded projection-valued measure on the Boolean algebra of open-closed sets, with the non-Archimedean orthogonality property f = sup Λ

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Baer subrings of the ring of linear transformations

Cited by 5 publications

References 3 publications

Distinguished Rings of Linear Transformations

Distinguished Rings of Linear Transformations

Isomorphisms and anti-isomorphisms of endomorphism rings of modules

On some classes of non-Archimedean operator algebras

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