Unit group of a direct product of rings ($U({\prod }_j{R}_j)={\prod }_{j}U(R_j)$)

A tuple in a direct product of rings is a unit exactly when each coordinate is a unit. Indeed, multiplying by an inverse tuple gives an inverse in each coordinate; conversely, the coordinate inverses form an inverse tuple. This identifies the unit group with the direct product of groups of the individual unit groups, for finite or infinite families of unital rings.