Properties Of Free Objects In The Category Of Riesz Spaces On Left R-modules

Riesz space acts on the left R-module to obtain the concept of Riesz space on the left R-module.Thus,the theoretical idea of modulus is partly derived from the theory of Riesz spaces and partly from the theory of left R-modules.Riesz space,also called vector lattice,which is a space of partial order vectors over real numbers,is introduced into Riesz space based on the related study of space in this paper,and the concept of Riesz space over left R-modules,also known as Riesz modes,is obtained.The free modules serve as the basis for the three basic classes of modules,namely,introjective modules,projective modules,and flat modules,and are an integral part of the module theory of rings.We use the category as a tool to generate the concept of the category of Riesz spaces on left R-modules,and introduce freedom into Riesz modules to obtain the concept of free Riesz modules,and prove the existence,uniqueness theorem of free Riesz modules,Riesz module M is a free Riesz module when and only when it has a basis,all basis equipotentials of free Riesz modules and every Riesz module M is a homomorphic image of some free Riesz module F.Also,based on the cleavability of the module-positive conjunction and the cleavability of the state projection,it is proved that the mapping of a Riesz module M to a free Riesz module F is cleavable if it is a full homomorphism.