We present the feasibility and limits of linear network coding in two-unicast-Z networks, i.e., two-unicast networks where one destination has apriori side-information of the unwanted, interfering message. We first present an approach to deriving achievability proofs based on the elementary techniques in commutative algebra. We demonstrate our approach by showing an alternate proof of the previous result of Wang et. al. related to feasibility of rate (1,1). 

We also demonstrate the limits of linear network coding in two-unicast-Z networks, by showing that vector linear codes outperform scalar linear codes,  non-linear codes outperform linear codes for two-unicast-Z networks. Because two-unicast-Z networks are special case of two-unicast networks, our results strengthen similar results obtained for two-unicast networks derived by Kamath et. al.