We develop and analyze new protocols to verify the correctness ofvarious computations on matrices over F[x], where F is a field. Theproperties we verify concern an F[x]-module and therefore cannotsimply rely on previously-developed linear algebra certificates whichwork only for vector spaces. Our protocols are interactivecertificates, often randomized, and featuring a constant number ofrounds of communication between the prover and verifier. We seek tominimize the communication cost so that the amount of data sent duringthe protocol is significantly smaller than the size of the resultbeing verified, which can be useful when combining protocols or insome multi-party settings. The main tools we use are reductions toexisting linear algebra certificates and a new protocol to verify thata given vector is in the F[x]-linear span of a given matrix.