Locally Split Homomorphisms Relative To A Sub module

In this article,for a positive integer n ,the concept of n-locally splithomomorphism relative to a sub module has been introduced andstudied,which will turn out to be most useful in the studying and providingcharacterizations of local projectivity, local-regularity in the sense of(Zelmanowitz, Field house and Ware) relative to a sub module.They presentgeneralization of projectivity and the three types of regularity which havebeen mentioned respectivily.Keywords: n-locally(T)-split homomorphisms,n-locally(T)-projective modules, n-(T)-regular modules,(T)-pure sub modules and Zelmanowitz (Field house) (T)-regularmodules.