On rings whose projective modules have the exchange property
Let R be an associative ring with identity. Modules over the ring R will be assumed unitary left R-modules. The Jacobson radical of the ring R will be denoted by J(R). A module M is said to have the exchange property if for every module A and any two decompositions \(A=M'\oplus N=\oplus_{i\in I}A_ i\), where M'\(\cong M\), there are submodules \(A_ i'\subset A_ i\) such that \(A=M'\oplus (\oplus_{i\in I}A_ i')\). M is said to have the finite exchange property if this condition is satisfied whenever the index set I is finite. A module M is called regular if every finitely generated submodule of M is a projective direct summand. A ring R is called a left P-exchange ring if every projective left R-module has the exchange property. The index of a nilpotent element \(x\in R\) is the least positive integer n such that \(x^ n=0\). R is said to have bounded index k, if k is the supremum of the indices of all nilpotent elements of R. R is called strongly \(\pi\)- regular, if for every \(a\in R\) there exist \(n\in {\mathbb{N}}\) and x,y\(\in R\) such that \(a^ n=a^{n+1}x=ya^{n+1}.\) Some of the results given by the author are: 1) Every regular projective module has the exchange property; 2) Every homomorphic image of a left P- exchange ring is a left P-exchange ring; 3) A ring R is a left P-exchange ring if and only if R/J(R) is a left P-exchange ring and J(R) is left T- nilpotent; 4) If M is a finitely generated left R-module, the following conditions are equivalent: (1) \(M^{(I)}\) has the exchange property for every set I, (2) \(End_ RM\) is a left P-exchange ring; 5) Let R be a ring all of whose idempotents are central. Then the following conditions are equivalent: (1) R is a left P-exchange ring. (2) The free left R- module \(R^{({\mathbb{N}})}\) has the finite exchange property. (3) R/J(R) is von Neumann regular and J(R) is left T-nilpotent; 6) Suppose that R has bounded index k. If all projective left R-modules have the finite exchange property, then R is strongly \(\pi\)-regular and every prime factor-ring of R is isomorphic to a matrix ring \(D^{n\times n}\) for some division ring D and some \(n\leq k\).
scientific article; zbMATH DE number 3973083
scientific article