 Solution

For congruence classes in $\mathbb{Z}/N\mathbb{Z}$, defineThese operations are independent of representatives. If $a^{\prime }=a+Nr$ and $b^{\prime }=b+Ns$, then $(a^{\prime }+b^{\prime })-(a+b)=N(r+s)$ and $a^{\prime }{b}^{\prime }-ab=N(rb+sa+Nrs)$. Thus changing representatives leaves both results in the same congruence class. The usual integer laws descend to make this a commutative ring with zero $[0]$ and identity $[1]$.