Definition:Degree of Polynomial/Ring

Definition
Let $R$ be a ring.

Let $S$ be a subring of $R$.

Let $x \in R$.

Let $\displaystyle P = \sum_{j \mathop = 0}^n \left({r_j \circ x^j}\right) = r_0 + r_1 x + \cdots + r_n x^n$ be a polynomial over $S$ in $x$ such that $r_n \ne 0$.

Then the degree of $P$ is $n$.

The degree of $P$ can be denoted $\deg \left({P}\right)$ or $\partial P$.