A Real Nullstellensatz for free modules

Let AA be the algebra of all n×nn×n matrices with entries from R[x1,…,xd]R[x1,…,xd] and let G1,…,Gm,F∈AG1,…,Gm,F∈A. We will show that F(a)v=0F(a)v=0 for every a∈Rda∈Rd and v∈Rnv∈Rn such that Gi(a)v=0Gi(a)v=0 for all i if and only if F   belongs to the smallest real left ideal of AA which contains G1,…,GmG1,…,Gm. Here a left ideal J   of AA is real if for every H1,…,Hk∈AH1,…,Hk∈A such that H1TH1+⋯+HkTHk∈J+JT we have that H1,…,Hk∈JH1,…,Hk∈J. We call this result the one-sided Real Nullstellensatz for matrix polynomials. We first prove by induction on n   that it holds when G1,…,Gm,FG1,…,Gm,F have zeros everywhere except in the first row. This auxiliary result can be formulated as a Real Nullstellensatz for the free module R[x1,…,xd]nR[x1,…,xd]n.