Symmetric closure in modules and rings

Abstract

We introduce both for modules and rings classes of elements that are strongly connected to commutativity classes as defined in Abdi, M., Leroy, A. G. (2021). Graphs of commutatively closed sets. Linear Multilinear Algebra. 70(21):6965–6977 and Alghazzawi, D., Leroy, A. G. Commutatively closed sets in rings. Commun. Algebra. 47(4):1629–1641. We define a graph structure on the classes leading to a notion of distance for elements of a class. Examples are presented all along the paper, showing the interest of the notions.