Definition:Degree of Polynomial/Ring

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Let $\struct {R, +, \circ}$ be a ring.

Let $\struct {S, +, \circ}$ be a subring of $R$.

Let $x \in R$.

Let $\ds P = \sum_{j \mathop = 0}^n \paren {r_j \circ x^j} = r_0 + r_1 \circ x + \cdots + r_n \circ x^n$ 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 $\map \deg P$ or $\partial P$.