Resumen de Some characterizations of regular modules

Goro Azumaya

• Let M be a left module over a ring R. M is called a Zelmanowitz-regular module if for each x Î M there exists a homomorphism F: M ? R such that f(x) = x. Let Q be a left R-module and h: Q ? M a homomorphism. We call h locally split if for every x Î M there exists a homomorphism g: M ? Q such that h(g(x)) = x. M is called locally projective if every epimorphism onto M is locally split. We prove that the following conditions are equivalent:

  (1) M is Zelmanowitz-regular.

  (2) every homomorphism into M is locally split.

  (3) M is locally projective and every cyclic submodule of M is a direct summand of M.