Let $R$ be a commutative ring with identity element, $U(R)$ its group of units and $J(R)$ its Jacobson radical. Recall that a ring $R$ is a $JU$-ring (resp. a weakly $JU$-ring) if $U(R)=1+J(R)$ (resp. $U(R)=\pm 1+J(R)$). In this note we investigate the properties of these notions in different contexts of commutative rings. Precisely, we study the transfer of the notions of $JU$-rings, $WJU$-rings, $UU$-rings, $WUU$-rings, and more to trivial ring extensions, amalgamations of rings and pullbacks. Our aim is to provide new classes of commutative rings satisfying these properties. Examples illustrating the limits and scopes of our results are provided.