Several properties of the inverse along an element are studied in the context of unitary rings. New characterizations of the existence of this inverse are proved. Moreover, the set of all invertible elements along a fixed element is fully described. Furthermore, commuting inverses along an element are characterized. The special cases of the group inverse, the (generalized) Drazin inverse and the Moore-Penrose inverse (in rings with involutions) are also considered.

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En la presente investigación, el objetivo principal fue realizar un control de longitudes de onda de luz UV, en un prototipo, analizando su óptica, cuyo referencial es el espectrofotómetro Perkin Elmer 781, usado en análisis de descontaminación de aguas surfactantes y de fotodegradación, para lo cual se ha rediseñado el generador, determinando el rango de frecuencias, disminuyendo el impacto ambiental, mediante la implementación de HMI, cuya metodología aplicada, para efectuar el proceso de automatización cuyo enfoque aplicativo se determina a través de la foto de graduación y descontaminación del agua, presentando un rediseño para la generación de longitudes de onda UV dentro de la parte eléctrica y electrónica a través de la interpolación de Newton disminuyendo el error al (0.05%) mínimos cuadrados de (3.32%) optimizando el control de motores con una conexión en HMI, mediante la cual se visualiza la longitud de onda generada facilitando la navegación y el control de los motores otorgando al operario la interacción directa con el equipo obteniendo como resultado a la mejora del sistema original variando los sensores y actuadores para obtener longitudes de onda de (500 nm) a (700 nm) el cual permite desarrollar trabajos en la descontaminación de aguas y efectividad del sistema.

Generalized inverses, known today as Classical Generalized Inverses, were studied during the rst decades of the last century. Two important classical generalized inverses are the Moore-Penrose inverse (1955) and the Drazin inverse (1958). The Moore-Penrose inverse was originally dened for complex rectangular matrices. In turn, the Drazin inverse was studied, at rst, only for square matrices. It was in 1980 when Cline and Greville extended the case of square matrices to the case of rectangular matrices by considering a weight rectangular matrix. Throughout the entire past century there appeared dierent properties, characterizations and applications of these types of generalized inverses.This last decade gave rise to new notions of generalized inverses. The rst of these new notions is known as the core inverse. Core inverses were introduced in 2010 by Baksalary and Trenkler. Their work had a wide repercussion in the mathematical community due to the simplicity of its denition and its application in the solution of some linear systems with restrictions. The core inverse further gain in interest due to their connection to the Bott-Dun inverse. There is a large body of work on the core inverse, including extensions to more general sets such as the algebra of bounded linear operators on Hilbert spaces and/or abstract rings. v