Proposition (Kernel of an Algebra Homomorphism): Let I be a two-sided ideal in the free monoid ring K, and let J be a two-sided ideal in the free monoid ring K. Moreover, let g_1,...,g_m be polynomials in K, and let phi: K/J-->K/I be a homomorphism of K-algebras defined by phi(y[i]+J)=g_i+I for i=1,...,m. We form the free monoid ring K, and let D be the diagonal ideal generated by the set {y[1]-g_1,...,y[m]-g_m}. Then we have ker(phi)=((D+J) intersets K)+I.
Corollary (Minimal Polynomial): Let phi: K[y]-->K/I be a K-algebra homomorphism given by phi(y)=g+I. Then g+I is algebraic over K if and only if ker(phi) is not zero. Moreover, if g+I is algebraic over K, then the unique monic generating polynomial of the ideal ker(phi) is the minimal polynomial of g+I over K.