Let $k$ be an [algebraically closed field](/page/Algebraically%20Closed%20Field), let $n \geq 0$, and set $R := k[x_0,\ldots,x_n]$ with its standard grading. For every set $S \subset R$ of homogeneous polynomials, define the projective zero locus

\begin{align*} V_+(S) := \{[a_0:\cdots:a_n] \in \mathbb{P}^n_k : f(a_0,\ldots,a_n)=0 \text{ for every } f \in S\}. \end{align*}

Equip $\mathbb{P}^n_k$ with the projective Zariski topology whose closed subsets are exactly the sets $V_+(S)$ for sets $S \subset R$ of homogeneous polynomials. For a [homogeneous polynomial](/page/Homogeneous%20Polynomial) $f \in R$, write $V_+(f) := V_+(\{f\})$. Let $X \subset \mathbb{P}^n_k$ be a nonempty projective algebraic set, meaning that $X=V_+(S)$ for some set $S \subset R$ of homogeneous polynomials, and equip $X$ with its Zariski [subspace topology](/page/Subspace%20Topology). Define its homogeneous vanishing ideal by

\begin{align*} I_+(X) := \bigoplus_{d \geq 0}\{f \in R_d : f(a_0,\ldots,a_n)=0 \text{ for every } [a_0:\cdots:a_n]\in X\}, \end{align*}

where $R_d$ is the $k$-[vector space](/page/Vector%20Space) of homogeneous degree-$d$ forms in $R$. Then $X$ is irreducible as a [topological space](/page/Topological%20Space) if and only if $I_+(X)$ is a prime ideal of $R$. In this case $I_+(X)$ is a proper homogeneous ideal and does not contain the irrelevant ideal $(x_0,\ldots,x_n)$.