DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-11

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The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution

Definition

Let $R$ be a commutative ring (Commutative ring). A function $a ⁣ : N \to R$ has finite support when there is $N \in N$ such that $a_{i} = 0$ for every $i > N$ . The polynomial ring $R [x]$ is the set of all finitely supported functions $a ⁣ : N \to R$ .

For $a, b \in R [x]$ and $n \in N$ , define

$(a + b)_{n} : = a_{n} + b_{n}, (a b)_{n} : = \sum_{i + j = n} a_{i} b_{j} .$

The convolution sum is a finite sum in the additive commutative monoid of $R$ (A finite sum in a commutative monoid indexed by an arbitrary finite set). Write $0$ for the zero sequence, $1$ for the sequence with coefficient $1_{R}$ at index $0$ and zero elsewhere, and $x$ for the sequence with coefficient $1_{R}$ at index $1$ and zero elsewhere. The coefficient sequence supported at $0$ with value $r$ is denoted again by $r$ , and a polynomial $a$ is written formally as $\sum_{i} a_{i} x^{i}$ .

The closure of these operations and the commutative-ring axioms are established by Coefficientwise sums and convolution products of finitely supported sequences are finitely supported and Polynomial convolution makes $R [x]$ a commutative ring containing $R$ as its constant subring.

Depends on

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…and 81 more results.

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Sources

Thomas W. Judson, Abstract Algebra: Theory and Applications, Chapter 17.1 (standard reference, not scraped)
Neil Donaldson, Math 120B Notes, Section 22 (standard reference, not scraped)