Multivariate polynomial ring

Definition

Let $R$ be a commutative unital ring. The $n$-variate polynomial ring over $R$ is defined as the ring of polynomials in $n$ symbols. If the $n$ symbols are $x_{1},x_{2},\ldots ,x_{n}$, then the polynomial ring is $x_{1},x_{2},\ldots ,x_{n}$.

The $n$-variate polynomial ring can be obtained by applying the polynomial ring operator $n$ times in succession.

When we simply say multivariate polynomial ring, we usually mean multivariate polynomial ring over a field.

Related notions

• Polynomial ring