How to change The Ring of the polynomial in Julia - julia

Question

Main
using AbstractAlgebra
include("./lib.jl");
S, (a,b,c) = PolynomialRing(QQ,["a","b","c"])
RR = AbstractAlgebra.RealField
s1 = S(b^2*a + c*a - 1)
s2 = S(c*a^2 + -b*a - c^4*a)
s3 = S(b*a + a + b)
poly_list = [s1,s2,s3]
resultant_system(poly_list,a)
function that has to change ring of the polynomial
using AbstractAlgebra;
function resultant_system(poly_list, eliminationVar)
parentRing = parent(poly_list[1])
# println("parentRing: ", parentRing, "\n")
# println("eliminationVar: ", eliminationVar)
arr = gens(parentRing)
LIST_SIZE = size(poly_list)[1]
parentVars = String[]
for i = gens(parentRing)
if i != eliminationVar
push!(parentVars, string(i))
end
end
println("parentVars: ", parentVars)
X, p = PolynomialRing(QQ, parentVars)
U, u = PolynomialRing(X, LIST_SIZE, "u")
V, v = PolynomialRing(U, LIST_SIZE, "v")
R, a = PolynomialRing(V, string(eliminationVar))
println("R:", R)
u_list = gens(U)
v_list = gens(V)
# poly_list[1] = b^2*a + c*a - 1 BUT
println(R(poly_list[1])) # this line not working
println(R(b^2*a + c*a - 1)) # this line works
end
I need to convert Multivariate Polynomial Ring in a, b, c over Rationals to
Univariate Polynomial Ring in a over Multivariate Polynomial Ring in v1, v2, v3 over Multivariate Polynomial Ring in u1, u2, u3 over Multivariate Polynomial Ring in b, c over Rationals
# poly_list[1] = b^2*a + c*a - 1 BUT
println(R(poly_list[1])) # this line not working
println(R(b^2*a + c*a - 1)) # this line works