Building a comprehensive foundational text in associative algebra theory, this seminal work offers a rigorous treatment of associative and nonassociative algebras. A. Adrian Albert pioneered methods for handling cyclic and crossed product algebras, developed novel factorization techniques for division algebras of prime-power degree, and unified algebraic results to yield simpler, generalized solutions that reduced reliance on complex classical ideal theory. The book also linked algebraic theory to new applications, such as Riemann matrices and involution algebras, thereby bridging historic algebraic theory with contemporary advances and applications in geometry and analysis. This once out-of-print masterpiece is now accessible to a new generation of students and includes a comprehensive bibliography of related topics for researchers.