Discussion Of Some Properties Of The Complexity

In this thesis,we mainly investigated the projective modules and their dimensions. Meanwhile, we discussed their properties on complexity of modules and got some valuable results.At first, we introduced some lemmas and corollaries and their proofs. Meanwhile, we introduced the concepts of projective modules,projective dimensions, global dimensions and their properties. Such as lpd(?)= (?){lpd(Aj)} and let P∈PR M, suppose the sequence 0→A→P→B→0 is exact, then lpd(B)≤lpd(A)+1 etc. We also generalized the later result.Secondly, this thesis defines the complexity of modules in algebra and algebraic complexity,.And then we discussed the properties of projective dimensions on the complexity of modules and we got some relevant results.Finally, we introduced some successor activities we can do in this thesis. For example, we can consider the properties of rings'global dimension on algebraic complexity; And we also can construct an amount to characterize complexity as we use the functor Ext of Hom to characterize projective dimensions, then we can discuss their properties.