On the Splitting Ring of a Polynomial

Let f(Z) = Zⁿ ? a₁Z^n?1 + … + (?1)^n?1a_n?1Z + (?1)ⁿa_n be a monic polynomial with coefficients in a ring R with identity, not necessarily commutative. We study the ideal I_f of R[X₁,…, X_n] generated by σ_i(X₁,…, X_n) ? a_i, where σ₁,…, σ_n are the elementary symmetric polynomials, as well as the quotient ring R[X₁,…, X_n]/I_f.