Let [Formula: see text] be a commutative ring with identity and [Formula: see text] an ideal of [Formula: see text]. We introduce and study the [Formula: see text]-weak global dimension [Formula: see text] of the factor ring [Formula: see text]. Let [Formula: see text] be a [Formula: see text]-linked extension of [Formula: see text], and we also introduce the [Formula: see text]-weak global dimension [Formula: see text] of [Formula: see text]. We show that the ring [Formula: see text] with [Formula: see text] is exactly a field and the ring [Formula: see text] with [Formula: see text] is exactly a [Formula: see text]. As an application, we give an upper bound for the [Formula: see text]-weak global dimension of a Cartesian square [Formula: see text]. More precisely, if [Formula: see text] is [Formula: see text]-linked over [Formula: see text], then [Formula: see text]. Furthermore, for a Milnor square [Formula: see text], we obtain [Formula: see text].
