  An associative ring with 1 is said to be semilocal provided it is semisimple
artinian modulo its Jacobson radical, that is, modulo its Jacobson radical it
is isomorphic to a finite product of matrices over division rings.
  Modules with a semilocal endomorphism ring inherit some properties of
semisimple modules. For example, they can be decomposed into a finite direct
sum of indecomposable submodules, such decomposition is not unique but there
are only a finite number of different ones, they cancel from direct sums and
they satisfy the n-th root uniqueness property.
  A ring homomorphism is said to be local if it carries non-units to non-units.
Semilocal rings can be characterized as those rings having a local homomorphism
to a semisimple artinian ring.
  In this paper show that local homomorphisms appear frequently in module
theory. As a consequence it will follow that many interesting classes of
modules, as for example finitely presented modules over semilocal rings, have a
semilocal endomorphism ring.
