An Ideal-Theoretic Characterization of the Ring of All Linear Transformations

Endomorphism Rings of Reduced Complete Torsion-Free Modules over Complete Discrete Valuation Rings

“…Of course, £ induces a certain ring of ë£e-endomorphisms of the ë£ê-vector space ëË, again by right multiplication. This must be the full ê£é-endomorphism ring, according to the proof of Wolfson's theorem in [5]. It follows at once from Theorem 5.6 in [3] that a subring S of ER(H) is dense in £#(77) in the y-adic topology if and only if S induces the full 7v//>7v-endomorphism ring in HjpH.…”

“…By generalizing the Harrison-Matlis duality of [1] and [4] to the noncommutative case, this work simultaneously characterizes endomorphism rings of divisible torsion modules. The model for our main result was Wolfson's beautiful characterization of the ring of all linear transformations of a vector space over a division ring in [5]. The purpose of this note is to show how Wolfson's theorem can be used directly for the characterization of the endomorphism rings of these modules.…”

Rings which are generated by their units: a graph theoretical approach

“…The study of rings which are generated additively by their units seems to have arisen in 1953-1954 when Wolfson [13] and Zelinsky [14] proved, independently, that if V is a finite or infinite dimensional vector space over a division ring D, then every linear transformation is the sum of two nonsingular linear transformations unless dim V = 1 and . Der folgende Beitrag behandelt eine Strukturfrage zur Theorie endlicher kommutativer Ringe.…”

The characterization problem for endomorphism rings

“…Significant advances were made (for some particular classes 32 and J?) by Wolfson [20] in the 50s; and by Metelli and Salce [14] and Liebert [11,12,13] in the 70s. A more general result in this connection was obtained by Franzsen and Schultz [3,Theorem 3.2] in 1983: they provide a solution to the Characterization Problem for the class J[ of all the (locally) free i?-modules over rings R which satisfy the condition that each nonzero summand of a (locally) free .R-module is indecomposable if and only if it is isomorphic to R R. [2] Endomorphism rings 117…”