Definition:Monomial of Polynomial Ring/Multiple Variables/Definition 1

Definition

Let $R$ be a commutative ring with unity.

Let $I$ be a set.

Let $R \sqbrk {\family {X_i}_{i \mathop \in I} }$ be a polynomial ring in $I$ variables $\family {X_i}_{i \mathop \in I}$.

Let $y \in R \sqbrk {\family {X_i}_{i \mathop \in I} }$.

The element $y$ is a monomial of $R \sqbrk {\family {X_i}_{i \mathop \in I} }$ if and only if there exists a mapping $a: I \to \N$ with finite support such that:

$y = \ds \prod_{i \mathop \in I} X_i^{a_i}$

where:

$\prod$ denotes the product with finite support over $I$

$X_i^{a_i}$ denotes the $a_i$th power of $X_i$.

Also see

• Equivalence of Definitions of Monomial of Polynomial Ring in Multiple Variables