Symmetric rings were introduced by Lambek to unify sheaf representations of commutative rings and reduced rings. We continue the study of symmetric rings. In this paper, we define a general reduced ring (with or without identity), a general semicommutative ring (with or without identity) and a general symmetric ring (with or without identity), and identify maximal general symmetric subrings and maximal general semicommutative subrings of matrix ring over general reduced rings.

general symmetric ring, general semicommutative ring, matrix ring, general reduced ring.