Master of Arts (MA)
This thesis studies the conditions under which certain U (G)-modules have integral dimension. Specifically, the paper explores the circumstances under which certain U (G)-modules are free modules. Two main theorems are proven for this endeavor. The first theorem takes the matrix representation A of a submodule N of a free left module M over an arbitrary ring R. The theorem states that if A can be put into reduced row echelon form, then N and M/N are both free left R-modules. The second theorem takes the matrix representation B of an arbitrary left R-module homomorphism T. If B can be put into reduced column echelon form, then Ker( T) is a free left R-module as well. The results of these theorems are well-known if R is a field or division ring. The results are also probably known for an arbitrary ring R , but there does not seem to be a readily accessible source in the standard literature.
Dharia, Sejal K., "Integral and non-integral dimension of modules over non-commutative rings." (2009). Master's Theses. 3988.