Unit group of a direct product of rings
= Unit group of a direct product of rings
{title2=$U(\prod_jR_j)=\prod_jU(R_j)$}
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>.