Time Shifting Property
$ x[n-n_0] $, with ROC=R$ \rightarrow z^{-n_0}X(z) $, with ROC=R except for possible additional of origin and infinity.
Scaling in Z-domain
$ {z_0}^{n}x[n] $, with ROC=R$ \rightarrow X(\frac{z}{z_0}) $, with ROC=$ \frac{1}{R} $
Convolution
$ x_1[n]\star x_2[n] $, with ROC $ R_1 $ and $ R_2 $ $ \rightarrow X_1(z)X_2(z) $, with ROC containing at least $ R_1\cap R_2 $
Corollary,
$ x[n]\rightarrow $ LTI system $ h[n] $ $ \rightarrow X[n]\star h[n] $
$ X[n]\star h[n]=Z^{-1}(X(z)H(z)) $
H(z) is called the "System's Function".
