Let be a ring and let be the ring of polynomials in the central indeterminate We will always write the coefficients of an element of on the left. Now, if is commutative and in then for all This is not true if is noncommutative. For example, let and where Then and hence Thus if then but

**Remark 1**. Let be a ring. If in and if every coefficient of is in the center of then for all

*Proof*. Let and Then, since commutes with every we have

**Remark 2**. Let be a ring and Then if and only if for some

*Proof*. Let If then since commutes with we have

Conversely, if for some then

and thus

**Lemma 1**. Let be a division algebra with the center Suppose that in If and then Thus if then

*Proof*. Let and Then

**Lemma 2**. Let be a division algebra with the center Suppose that is algebraic over and is the minimal polynomial of over If is non-zero and for all then

*Proof*. Let and suppose that the lemma is false. Let be the smallest integer for which there exists a non-zero polynomial of degree such that and for all Note that since for every the polynomial has degree and for all we may assume that is monic. Let To get a contradiction, we are going to find a non-zero polynomial such that and for all We first note that, since is the minimal polynomial of over and we have So there exists such that Let be such that So Now we let

Then Hence and because Let Then, by Lemma 1,