Definition:Transcendental (Abstract Algebra)/Ring

Definition
Let $\left({R, +, \circ}\right)$ be a commutative ring with unity whose zero is $0_R$ and whose unity is $1_R$.

Let $\left({D, +, \circ}\right)$ be an integral subdomain of $R$.

Let $\alpha \in R$.

Then $\alpha$ is transcendental over $D$ iff:
 * $\displaystyle \forall n \in \Z_{\ge 0}: \sum_{k \mathop = 0}^n a_k \circ \alpha^k = 0_R \implies \forall k: 0 \le k \le n: a_k = 0_R$

That is, $\alpha$ is transcendental over $D$ iff the only way to express $0_R$ as a polynomial in $\alpha$ over $D$ is by the null polynomial.

Notation
For such an $\alpha$ transcendental over $D$, it is conventional to use the letter $X$.

Thus a ring of polynomials over $D in such a transcendental is therefore usually denoted $D \left[{X}\right]$.'' in such a transcendental is therefore usually denoted $D \left[{X}\right]$.

Also see
If $\alpha \in R$ is not transcendental over $D$ then it is algebraic over $D$.