Analogs of Jacobian conditions for subrings

We present a generalization of the Jacobian Conjecture for m polynomials in n variables: f1,…,fm∈k[x1,…,xn], where k is a field of characteristic zero and m∈{1,…,n}. We express the generalized Jacobian condition in terms of irreducible and square-free elements of the subalgebra k[f1,…,fm]. We also discuss obtained properties in a more general setting - for subrings of unique factorization domains.