Fitting ideals of high order universal modules

Working with Fitting ideals of a module in Commutative Algebra is one of the important tools in examining the structure of the module and determining the regularity of the ring. In addition, Fitting ideals are used to solve some problems of physics related to dynamic systems. The main purpose of this thesis is to examine the algebraic properties of the universal module defined on the tensor product, such as freeness, projective dimension, in case the Fitting ideals of universal modules are invertible and to apply these results on the samples. In the introduction part of the thesis, studies on literature related to Fitting ideals and universal modules are mentioned. In the second part of the thesis, basic definitions such as ideal, module, tensor product, and some related features are given. In the third chapter, basic definitions, theorems and properties of universal modules are given. Then, the projective dimensions of the universal modules are examined on the samples. In the fourth chapter, definition and properties of Fitting ideal of universal modules are given. In the fifth chapter, after giving the definition and properties of the invertible ideal, the results obtained in the case of Fitting ideals of the first and second order universal modules defined on the tensor product of the two rings are expressed with proofs. In the sixth chapter, by giving the definition and properties of the symmetric power module of a module, the projective dimensions of the first and second order universal module defined on the tensor product of the two rings are examined.