Basic examples and extensions of symmetric rings

Symmetric rings were introduced by Lambek to unify sheaf representations of commutative rings and reduced rings. We continue the study of symmetric rings, discussing basic examples and extensions. From any given reduced ring we first construct a nonreduced symmetric ring, and observe the form of the minimal noncommutative symmetric ring of order 16 up to isomorphism. We next show that polynomial rings over symmetric rings need not be symmetric and that classical right quotient rings of right Ore symmetric rings are symmetric. We also construct more examples of symmetric rings and counterexamples to several naturally raised situations in the process.