The structure of Moore Penrose rings are analyzed with precise pristine logic by examining the detailed pulchritude of these prolific algebraic systems in the engineering and physical science disciplines. The early portion of this literary exposition employs formal ring classification by dissecting MP1 (or Von Neumann regular) rings from a perspective which justifies the usefulness of Moore Penrose rings in the error analysis of statistical regression, or as an optimization mechanism using the method of least squares. Subsequent sections contain both local and global inspections of these multifaceted Moore Penrose rings: their characterizations by idempotents; their integral extensions and matrix ring embeddings; their modular qualities and representations as direct sums and summands, as well as their preservation under canonical ring homomorphisms. This concise and taxonomical treatment of Moore Penrose rings allows this introduction to be a handy platform for promoting their research potential-not to overlook their appeal for reinforcing basic ring theory concepts for the beginning contemporary algebra student.