Let A be a finite dimensional commutative and associative algebra with identity, over a field K. We assume also that A is generated by one element and consequently, isomorphic to a quotient algebra of the polynomial algebra K[X]. If A=K[a] and bi=fi(A), fi(X) ∊ K[X], 1≤i≤r we find necessary and sufficient conditions which should be satisfied by fi(X) in order that A = K[b1, …, br].

The result can be stated as a theorem about matrices. As a special case we obtain a recent result of Thompson [4].

In fact this last result was established earlier by Mirsky and Rado [3]. I am grateful to the referee for supplying this reference.