  We study (inhomogeneous) approximation for systems of linear forms using
integer points which satisfy additional primitivity constraints. The first
family of primitivity constraints we consider were introduced in 2015 by Dani,
Laurent, and Nogueira, and are associated to partitions of the coordinate
directions. Our results in this setting strengthen a theorem of Dani, Laurent,
and Nogueira, and address problems posed by those same authors. The second
primitivity constraints we consider are analogues of the coprimality required
in the higher-dimensional Duffin--Schaeffer conjecture, posed by Sprind\v{z}uk
in the 1970's and proved by Pollington and Vaughan in 1990. Here, with
attention restricted to systems of linear forms in at least three variables, we
prove a univariate inhomogeneous version of the Duffin--Schaeffer conjecture
for systems of linear forms, the multivariate homogeneous version of which was
stated by Beresnevich, Bernik, Dodson, and Velani in 2009 and recently proved
by the second author.
