Definition:Degree of Polynomial/Ring

Definition

Let $\left({R, +, \circ}\right)$ be a ring.

Let $\left({S, +, \circ}\right)$ be a subring of $R$.

Let $x \in R$.

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