Stable modules and a theorem of Camillo and Yu

In 1995, Camillo and Yu showed that an exchange ring has stable range 1 if and only if every regular element is unit-regular. An element m   in a module MR is called regular if (mλ)m=m(mλ)m=m for some λ∈hom(M,R)λ∈hom(M,R). In this paper we define stable modules and show that if M has the finite exchange property then M   is stable if and only if, for every regular element m∈Mm∈M, (mγ)m=m(mγ)m=m where γ:M→Rγ:M→R is epic (and we say that m   is unit-regular). Such modules are called regular-stable. It is shown that RR is regular-stable if and only if R has internal cancellation. To simplify the exposition, many arguments are formulated in an arbitrary Morita context.