(a) If R is a ring. then so is R^op. where R^op is defined as follows. The underlying set of R^op is precisely Rand addition in R^op coincides with addition in R. Multiplication in R^op, denoted ⁰ , is defined by a ^O b = ba. where ba is the product in R. R^op is called the opposite ring of R.

(b) R has an identity if and only if R^op does.

(c) R is a division ring if and only if R^op is.

(d) (R^op)^op) = R.

(e) If S is a ring, then R ≅ S if and only if R^op ≅ S^op