Opposite ring

Let $R$ be a ring . The opposite ring $R^{\mathrm{op}}$ is defined to have the same underlying additive group as $R$, but with multiplication given by

The construction is functorial and reverses multiplication: a homomorphism $R\to S$ induces $R^{\mathrm{op}}\to S^{\mathrm{op}}$. The identity map identifies $R$ with $R^{\mathrm{op}}$ exactly when $R$ is commutative , and the center is unchanged by passing to the opposite ring.

Examples:

• If $R$ is commutative (e.g. $\mathbb{Z}$), then $R^{\mathrm{op}}=R$ as rings.
• For the matrix ring $M_n(R)$, transposition gives an isomorphism $(M_n(R))^{\mathrm{op}}\cong M_n(R^{\mathrm{op}})$.
• If $R$ is the ring of upper-triangular $2\times 2$ matrices over a field, then $R^{\mathrm{op}}$ is naturally isomorphic to the ring of lower-triangular matrices.