The purpose of this dissertation is to study pseudo-Riemannian algebras, which are algebras with pseudo-Riemannian non-degenerate symmetric bilinear forms. The paper([1]), the authors Zhiqi Chen, Ke Liang, and Fuhai Zhu find that pseudo-Riemannian algebras whose left centers are isotropic play a key role and show that the decomposition of pseudo-Riemannian algebras whose left centers are isotropic into indecomposable non-degenerate ideals is unique up to a special automorphism. Furthermore, if the left center equals the center, the orthogonal decomposition of any pseudo-Riemannian algebra into ...