opposite ring

Look at a noncommutative ring in a mirror that swaps the order of every product. Addition is untouched, but wherever the ring multiplied a then b, the mirror image multiplies b then a. The reflected ring is the opposite ring. For a commutative ring the mirror shows nothing new, but for a genuinely noncommutative ring the opposite can be a subtly different beast — sometimes isomorphic to the original, sometimes not.

Precisely, for a ring R the opposite ring R-op has the same underlying additive group and a new multiplication a * b (in R-op) defined as b * a (in R). It is again a ring with the same identity, and (R-op)-op = R. A ring is commutative exactly when R = R-op as rings with the given multiplication. The construction is functorial and reverses the direction of homomorphisms in a controlled way.

The opposite ring is the bridge between left and right. A right R-module is precisely a left R-op-module, which is why one rarely needs a separate theory for each side. For matrix rings, transposition gives an isomorphism M_n(R)-op ≅ M_n(R-op), and for a field k the algebra M_n(k) is isomorphic to its own opposite via transpose. The endomorphism ring of R as a right module over itself is R-op, a clean conceptual appearance.

In the Brauer group the opposite supplies inverses: the class of A-op is the inverse of the class of A, because A tensor A-op is a full matrix algebra over the center. Whether a division algebra is isomorphic to its opposite is a real question — for instance it detects whether the algebra carries an anti-automorphism, which is the algebraic shadow of an involution like complex conjugation or quaternion conjugation.