Product ring in the context of Isomorphism

In mathematics, a product of rings or direct product of rings is a ring that is formed by the Cartesian product of the underlying sets of several rings (possibly an infinity), equipped with componentwise operations. It is a direct product in the category of rings.

Since direct products are defined up to an isomorphism, one says colloquially that a ring is the product of some rings if it is isomorphic to the direct product of these rings. For example, the Chinese remainder theorem may be stated as: if m and n are coprime integers, the quotient ring ${\displaystyle \mathbb {Z} /mn\mathbb {Z} }$ is the product of ${\displaystyle \mathbb {Z} /m\mathbb {Z} }$ and ${\displaystyle \mathbb {Z} /n\mathbb {Z} .}$