The endomorphism ring of a locally free module

Abstract

We identify a large class of rings over which locally free modules are determined by their endomorphism rings. We characterize these endomorphism rings and consider under what circumstances the conditions on the locally free modules can be relaxed, for example by requiring that only one of the rings need be in the special class, or by replacing "free' by "projective".

Isomorphisms Between Endomorphism Rings of Modules

“…If (F, A) contains a point, and e is an associated idempotent, the bimodule (F, A, E) is isomorphic to the bimodule (eEe, eE, E) in the sense that: (1) there is a semi-linear isomorphism (p, y) of the left modules (F, A) and (eEe, eE). (2) y is an E-isomorphism of the right modules (A, E) and (eE, E).…”

Isomorphisms between endomorphism rings of modules

“…Its proof is implicit in section 4 of [2]. See also [ Conversely, for any point of A, let e be an associated idempotent.…”

The characterization problem for endomorphism rings

“…Then the fact that Ae i is a finite direct sum of indecomposable modules, by the finiteness condition on e i , implies that the same happens to R. REMARK 4.4. In [3], a ring R is said to be an /F-ring when every nonzero summand of a free .R-module is indecomposable if and only if it is isomorphic to R R. Corollary 4.3 can be obviously applied to /F-rings: if R is such, then one can choose the family {e tJ } in the natural manner of the proof of Theorem 4.2, and then F is the finite topology on A = End( R F), A Q contains all the finite idempotents and each e t is primitive. Thus it is easy to obtain from this [13, Theorem 3.1], for if the conditions therein hold, then the direct sum ® 7 .Ee, (in the notation of [13]) may be written as © EUj for primitive idempotents M and hence one can see that (b) PROOF.…”

“…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…”