SEMI-COMPLEMENTED GROUP RINGS

Within mathematics, the field of algebra focuses primarily on preserving symmetry and structure within a set. This thesis focuses on a given group G over a (commutative) ring R, referred to as the group ring R[G]. Elements within R[G] are finite formal sums of elements of G with coefficients from R. A class of reduced rings that generalizes integral domains is that of complemented rings. We will discuss new subcategories of complemented rings: semi-complemented and almost complemented rings. Since R[G] is a ring itself, we would like to find when R[G] is complemented or perhaps only semi or almost complemented. We will discuss small examples such as Z2[D3] and Zk[Cn].