Replacing units by unipotents

A unit u is called unipotent if u − 1 is nilpotent. As special classes of stable range one elements, replacing in the definitions units by unipotents, strong left (or right) stable range one elements, unipotent left (or right) stable range elements, strong left (or right) unipotent stable range one elements and unipotent-regular elements are defined. All these type of elements (and the corresponding rings, for some of these) are studied with special emphasis to matrices.