Central \(\sigma\)-Symmetric Rings and Their Structural Properties

For an associative ring \(R\) with identity and an endomorphism \(\sigma\), the central \(\sigma\)-symmetric condition requires that \(a b c=0\) imply both \(a c \sigma(b)\) and \(\sigma(b) a c\) belong to the centre of \(R\). This condition was considered by Dutta and Singh (2022) in connection with \(\sigma\)-symmetric rings. We examine the distinction between its one-sided forms using explicit triangular-matrix examples, describe its relationship with \(\sigma\)-compatibility, symmetric rings, and skew central reversibility, and establish structural criteria for subrings, isomorphic images, quotients, diagonal matrix rings, central localizations, and direct products. The distinction between unital and nonunital endomorphisms is made explicit. In particular, \(\sigma\)-compatibility together with ordinary symmetry yields the stronger \(\sigma\)-symmetric property, whereas central \(\sigma\)-symmetry need not imply \(\sigma\)-symmetry. The arguments describe the precise hypotheses under which these implications and constructions hold.