Affine varieties are the basic geometric objects of classical algebraic geometry. They arise from a simple but powerful reversal: instead of studying a polynomial by its coefficients, study the set of points where many polynomials vanish at once. This turns algebra in a polynomial ring into geometry in affine space, and it is the entry point for ideas such as [Polynomial Ring](/page/Polynomial%20Ring), [Ideal](/page/Ideal), Zariski Topology, Coordinate Ring, and Scheme. The definition is flexible enough to describe familiar curves and surfaces, but rigid enough to remember algebraic structure. A line, a parabola, a plane conic, a finite set of points, and a hypersurface can all be described by polynomial equations. The reason affine varieties matter is that many geometric questions become algebraic questions about ideals, quotient rings, and homomorphisms. Over an [algebraically closed field](/page/Algebraically%20Closed%20Field), affine varieties are tightly controlled by the Hilbert Nullstellensatz. Over a non-algebraically closed field, there is a useful distinction between equations with coefficients in $k$ and points whose coordinates lie in $k$; that distinction is one of the first signs that algebraic geometry must eventually separate schemes, rational points, and base change. ## Definition Affine varieties are the basic geometric objects studied through polynomial equations, so we begin by isolating the ambient algebraic setting in which those equations live.. The definition below packages the ambient affine space and the common-zero-locus condition together; the separate definitions of [Affine Space](/page/Affine%20Space) and vanishing set then isolate the notation used throughout the rest of the page. [definition: Affine Variety] Let $k$ be a field and let $n \in \mathbb{N}$. An affine variety over $k$ is a subset $X \subset \mathbb{A}^n_k := k^n$ for which there exists a set $S \subset k[x_1, \ldots, x_n]$ such that \begin{align*} X = \{p \in \mathbb{A}^n_k : f(p)=0 \text{ for every } f \in S\}. \end{align*} [/definition] Polynomial equations need a fixed universe of points before their zero sets can be discussed. The same symbols $x_1, \ldots, x_n$ can describe coordinates on many different ambient spaces, depending on the base field and the number of coordinates. Affine space supplies that ambient universe, so later notation can distinguish the space of all possible coordinate tuples from the particular solution sets cut out inside it. [definition: Affine Space] Let $k$ be a field and let $n \in \mathbb{N}$. The affine $n$-space over $k$ is the set \begin{align*} \mathbb{A}^n_k := k^n. \end{align*} A point $p \in \mathbb{A}^n_k$ is written $p=(a_1, \ldots, a_n)$ with $a_i \in k$. [/definition] Once the ambient space is fixed, the next recurring operation is to pass from equations to their simultaneous solutions. Isolating this operation lets us compare different equation lists, describe intersections and unions later, and speak about the zero locus of an ideal without restating the evaluation condition each time. [definition: Vanishing Set] Let $k$ be a field, let $n \in \mathbb{N}$, and let $S \subset k[x_1, \ldots, x_n]$. The vanishing set of $S$ in $\mathbb{A}^n_k$ is \begin{align*} V(S) := \{p \in \mathbb{A}^n_k : f(p)=0 \text{ for every } f \in S\}. \end{align*} [/definition] Many authors reserve the word variety for irreducible algebraic sets over an algebraically closed field. On this page, an affine variety means any common zero locus $V(S)$, including reducible and empty examples. When irreducibility matters, it is stated as an additional condition. For example, in $\mathbb{A}^2_k$ the equation $y=0$ defines \begin{align*} V(y)=\{(a,0):a\in k\}, \end{align*} the affine line sitting inside the affine plane. This small example is the template: a polynomial equation is algebra, while its vanishing set is the geometric object. A subset of affine space also determines polynomials: namely, the polynomials that vanish on every point of the subset. To compare different equation lists for the same variety, we need this reverse construction from geometry back to algebra. [definition: Vanishing Ideal] Let $k$ be a field, let $n \in \mathbb{N}$, and let $X \subset \mathbb{A}^n_k$. The vanishing ideal of $X$ is \begin{align*} I(X) := \{f \in k[x_1, \ldots, x_n] : f(p)=0 \text{ for every } p \in X\}. \end{align*} [/definition] The [vanishing ideal is an ideal](/theorems/9404) in the polynomial ring. It records all polynomial equations satisfied by $X$, not just one chosen list of equations. We need the coordinate ring because polynomial functions on $X$ should identify ambient polynomials that agree at every point of $X$. Quotienting by $I(X)$ performs exactly that identification and turns the geometry of $X$ into a commutative algebra object. [definition: Coordinate Ring of an Affine Variety] Let $k$ be a field and let $X \subset \mathbb{A}^n_k$ be an affine variety. The coordinate ring of $X$ is the [quotient ring](/page/Quotient%20Ring) \begin{align*} k[X] := k[x_1, \ldots, x_n]/I(X). \end{align*} [/definition] The coordinate ring is the algebra of polynomial functions on the chosen set of $k$-points $X$. Over algebraically closed fields this point-set construction matches the usual classical dictionary especially well; over finite or non-algebraically closed fields, $I(X)$ may remember only what happens on the displayed $k$-rational points, so the field of definition must remain part of the data. It is still the main reason affine varieties are algebraically tractable: maps between affine varieties are encoded by homomorphisms between coordinate rings in the opposite direction. ## Equivalent Characterisations The definition permits an arbitrary subset $S$ of the polynomial ring, but the geometry does not depend on the particular generating set. Since a point annihilating every polynomial in $S$ also annihilates every polynomial combination of elements of $S$, the common zero locus should depend only on the generated ideal. [quotetheorem:9517] The algebraic object naturally attached to equations is therefore an ideal, not a bare list of polynomials. To make the definition usable in calculations, we also need to know whether infinitely many equations can always be replaced by finitely many. [quotetheorem:9518] This finiteness result is what makes the definition of affine variety usable rather than merely formal. A subset could be described as the common zero locus of many equations, but over a polynomial ring in finitely many variables the [Hilbert basis theorem](/theorems/860) gives a finite list with the same zero set. Geometrically, this means an affine variety has a finite presentation by equations: one can compute with a finite system, compare examples, and pass from geometry to ideals without carrying an infinite amount of defining data. The finite list is not unique. Different finite systems of equations may still cut out the same set of points. The next algebraic condition isolates ideals that have already forgotten the power information invisible to common zero loci. [definition: Radical Ideal] Let $R$ be a commutative ring and let $I \trianglelefteq R$. The ideal $I$ is radical if for every $f \in R$ and every $m \in \mathbb{N}$, \begin{align*} f^m \in I \implies f \in I. \end{align*} [/definition] Radical ideals are the right ideals for classical affine varieties because a polynomial and a power of that polynomial vanish on exactly the same set of points. The precise correspondence requires an algebraically closed field of points. In the quoted theorem, $\Omega$ denotes that algebraically closed field, and $k$ is the coefficient field for the equations being evaluated on $\Omega^n$. Thus a "$k$-algebraic subset of $\Omega^n$" means a zero locus in $\Omega^n$ cut out by polynomials with coefficients in $k$. In the main classical case used on this page, take $k=\Omega$; then $\Omega^n$ is the affine space $\mathbb{A}^n_\Omega$ and $k[T_1,\ldots,T_n]$ is just the ambient polynomial ring over the same algebraically closed field. [quotetheorem:2941] The [Nullstellensatz correspondence](/theorems/2870) explains why geometric containment becomes ideal inclusion in the opposite direction. It also suggests that decomposing a variety into pieces should be reflected by a corresponding algebraic decomposition. To study components, dimension, and prime ideals, we need a notion of an affine variety made from one algebraic piece. Reducible examples occur often, so the indecomposable case deserves its own name. [definition: Irreducible Affine Variety] Let $k$ be a field and let $X \subset \mathbb{A}^n_k$ be an affine variety. The variety $X$ is irreducible if $X \ne \varnothing$ and whenever $Y,Z \subset X$ are affine varieties with \begin{align*} X = Y \cup Z, \end{align*} then $X=Y$ or $X=Z$. [/definition] Irreducibility is the algebraic-geometric analogue of being made from one piece. To make this topological-looking idea computable, we relate it to multiplication in the coordinate ring and primeness of the vanishing ideal. As in the [Nullstellensatz correspondence](/theorems/2941) above, the quoted theorem uses $\Omega$ for an algebraically closed base field; thus $\Omega^n$ means $\mathbb{A}^n_{\Omega}$, and $\Omega[T_1,\ldots,T_n]$ is the polynomial ring in the ambient coordinates over that field. [quotetheorem:2942] This theorem is one of the first precise signs of the dictionary between geometry and algebra: unions correspond to intersections of ideals, while irreducibility corresponds to primeness. In practice it lets one test a geometric question by computing in the coordinate ring: if $I(X)$ is prime, then two regular functions whose product vanishes on $X$ cannot both vanish only on complementary pieces. For example, the union of the two coordinate axes in the affine plane has equation $xy=0$ and is reducible; algebraically, the ideal generated by $xy$ is not prime because $xy$ lies in it while neither $x$ nor $y$ does. By contrast, a curve whose coordinate ring is an [integral domain](/page/Integral%20Domain) passes this test and has no decomposition into two proper affine subvarieties. The criterion must be applied to the vanishing ideal $I(X)$, not blindly to an arbitrary list of equations defining the same set. Extra nilpotent or power information in a non-radical equation ideal can obscure the geometry. The algebraically closed hypothesis is also part of the classical form of the dictionary: over non-algebraically closed fields, statements about rational points can miss components that appear after extending scalars. ## Standard Examples The simplest examples show that affine varieties include the usual algebraic curves from coordinate geometry. They also show how the same geometric set can have many different equations. [example: The Affine Line] Let $k$ be an infinite field. The affine line $\mathbb{A}^1_k$ is an affine variety because the zero polynomial vanishes at every point: \begin{align*} V(0)=\{a \in \mathbb{A}^1_k : 0(a)=0\}=\mathbb{A}^1_k. \end{align*} We compute its vanishing ideal. By definition, \begin{align*} I(\mathbb{A}^1_k)=\{f \in k[x] : f(a)=0 \text{ for every } a \in k\}. \end{align*} The zero polynomial belongs to this set. Conversely, suppose $f \in k[x]$ is nonzero and has degree $d$. A nonzero polynomial of degree $d$ over a field has at most $d$ roots: indeed, if $f(a)=0$, then the [factor theorem](/theorems/3235) gives $f=(x-a)g$ with $\deg(g)=d-1$, and repeating this argument for distinct roots removes one linear factor each time. Since $k$ is infinite, there are more than $d$ elements of $k$, so $f$ cannot vanish at every element of $k$. Hence the only polynomial vanishing on all of $\mathbb{A}^1_k$ is $0$, and therefore \begin{align*} I(\mathbb{A}^1_k)=(0). \end{align*} The coordinate ring is then \begin{align*} k[\mathbb{A}^1_k]=k[x]/I(\mathbb{A}^1_k)=k[x]/(0)=k[x]. \end{align*} Thus affine space itself is a variety, not merely an ambient container. The infinite-field hypothesis matters: if $k=\mathbb{F}_q$, then for $a=0$ one has $a^q-a=0$, while for $a \ne 0$ the group $\mathbb{F}_q^\times$ has order $q-1$, so $a^{q-1}=1$ and \begin{align*} a^q-a=a(a^{q-1}-1)=a(1-1)=0. \end{align*} Therefore $x^q-x$ vanishes on every element of $\mathbb{F}_q$, so the vanishing ideal of the set of $\mathbb{F}_q$-points of $\mathbb{A}^1_{\mathbb{F}_q}$ contains the nonzero polynomial $x^q-x$ and is not $(0)$. [/example] The next example is the standard model for hypersurfaces: a single equation cuts out a codimension-one object when the polynomial is not degenerate. [example: A Plane Parabola] Let $k$ be an infinite field and let \begin{align*} X=V(y-x^2)\subset \mathbb{A}^2_k. \end{align*} For every $t\in k$, \begin{align*} (y-x^2)(t,t^2)=t^2-t^2=0, \end{align*} so $\varphi:\mathbb{A}^1_k\to X$ defined by $\varphi(t)=(t,t^2)$ has image in $X$. Conversely, if $(a,b)\in X$, then \begin{align*} b-a^2=0, \end{align*} so $b=a^2$ and therefore $(a,b)=\varphi(a)$. Thus $X=\{(t,t^2):t\in k\}$. We now compute $I(X)$. Since $y-x^2$ vanishes on every point of $X$, we have \begin{align*} (y-x^2)\subset I(X). \end{align*} For the reverse inclusion, take $f\in I(X)$. Define the $k$-algebra homomorphism \begin{align*} \theta:k[x,y]\to k[t] \end{align*} by $\theta(x)=t$ and $\theta(y)=t^2$. Then \begin{align*} \theta(f)=f(t,t^2). \end{align*} Because $f\in I(X)$, for every $a\in k$ one has \begin{align*} \theta(f)(a)=f(a,a^2)=0. \end{align*} A nonzero polynomial in one variable over a field has only finitely many roots, while $k$ is infinite, so $\theta(f)=0$ in $k[t]$. It remains to identify $\ker(\theta)$. For every monomial $x^i y^j$, \begin{align*} x^i y^j-x^{i+2j}=x^i\bigl(y^j-(x^2)^j\bigr). \end{align*} Using the identity $u^j-v^j=(u-v)(u^{j-1}+u^{j-2}v+\cdots+v^{j-1})$, the difference $y^j-(x^2)^j$ is divisible by $y-x^2$. Hence each monomial $x^i y^j$ is congruent to $x^{i+2j}$ modulo $(y-x^2)$. Therefore every polynomial $f\in k[x,y]$ is congruent modulo $(y-x^2)$ to the one-variable polynomial $\theta(f)$, with $t$ replaced by $x$. Since $\theta(f)=0$, this congruence gives $f\in (y-x^2)$. Thus \begin{align*} I(X)=(y-x^2). \end{align*} The coordinate ring is therefore \begin{align*} k[X]=k[x,y]/I(X)=k[x,y]/(y-x^2). \end{align*} The map $\theta$ is surjective because $\theta(x)=t$, and its kernel is $(y-x^2)$, so it induces an isomorphism \begin{align*} k[x,y]/(y-x^2)\cong k[t]. \end{align*} Under this isomorphism, the class of $x$ maps to $t$ and the class of $y$ maps to $t^2$. Thus the parabola is curved as a subset of $\mathbb{A}^2_k$, but its polynomial functions are algebraically the same as polynomial functions on the affine line. Over a finite field, extra polynomials can vanish on all $k$-points, so this coordinate-ring computation uses the infinite-field hypothesis. [/example] Finite sets are also affine varieties over many fields, but the equations that define them reveal how the base field matters. This example is a useful boundary case because it is zero-dimensional but still algebraic. [example: A Finite Set of Points] Let $k$ be a field and let $a_1,\ldots,a_m\in k$ be distinct. Put \begin{align*} P(x)=(x-a_1)\cdots(x-a_m). \end{align*} For each $j$, the factor $x-a_j$ vanishes at $x=a_j$, so \begin{align*} P(a_j)=(a_j-a_1)\cdots(a_j-a_j)\cdots(a_j-a_m)=0. \end{align*} Thus every point $a_j$ lies in $V(P)$. Conversely, if $b\in V(P)$, then \begin{align*} (b-a_1)\cdots(b-a_m)=0. \end{align*} A field has no zero divisors, so at least one factor is zero. Hence $b-a_j=0$ for some $j$, and therefore $b=a_j$. Thus \begin{align*} V(P)=\{a_1,\ldots,a_m\}=X. \end{align*} So $X$ is an affine variety in $\mathbb{A}^1_k$. We compute its vanishing ideal. Since $P(a_j)=0$ for every $j$, every multiple of $P$ vanishes on $X$, so \begin{align*} (P)\subset I(X). \end{align*} For the reverse inclusion, let $f\in I(X)$. Since $f(a_1)=0$, the factor theorem gives \begin{align*} f=(x-a_1)g_1 \end{align*} for some $g_1\in k[x]$. For $j\geq 2$, evaluating at $a_j$ gives \begin{align*} 0=f(a_j)=(a_j-a_1)g_1(a_j). \end{align*} Because $a_j-a_1\neq 0$ in the field $k$, we get $g_1(a_j)=0$. Applying the same argument successively to $g_1$ at $a_2$, then to the next quotient at $a_3$, and so on, gives \begin{align*} f=(x-a_1)(x-a_2)\cdots(x-a_m)h \end{align*} for some $h\in k[x]$. Hence $f\in(P)$, so \begin{align*} I(X)=(P). \end{align*} The coordinate ring is therefore \begin{align*} k[X]=k[x]/I(X)=k[x]/((x-a_1)\cdots(x-a_m)). \end{align*} The factors are pairwise comaximal: if $i\neq j$, then \begin{align*} (x-a_i)-(x-a_j)=a_j-a_i. \end{align*} Since $a_j-a_i\in k^\times$, this identity implies $1\in (x-a_i)+(x-a_j)$. By the *[Chinese remainder theorem](/theorems/734)*, \begin{align*} k[x]/((x-a_1)\cdots(x-a_m))\cong \prod_{j=1}^m k[x]/(x-a_j). \end{align*} For each $j$, evaluation at $a_j$ induces an isomorphism \begin{align*} k[x]/(x-a_j)\cong k. \end{align*} Combining these identifications gives \begin{align*} k[X]\cong k^m. \end{align*} Thus a finite set of $m$ distinct $k$-points has coordinate ring equal to a product of $m$ copies of $k$, one copy for each point. If $k$ is algebraically closed, every proper affine variety in $\mathbb{A}^1_k$ is finite. Indeed, if $Y=V(S)\neq \mathbb{A}^1_k$, choose $b\notin Y$. Then some $f\in S$ satisfies $f(b)\neq 0$, so $f$ is nonzero and $Y\subset V(f)$. Since $k$ is algebraically closed, $f$ factors as \begin{align*} f=c(x-b_1)\cdots(x-b_r) \end{align*} with $c\in k^\times$ and $b_1,\ldots,b_r\in k$, so $V(f)\subset\{b_1,\ldots,b_r\}$. Hence $Y$ is a finite set of points, and the construction above writes it as the zero set of the product of the corresponding linear factors. [/example] Not every familiar-looking equation behaves the same over every field. The next example shows why the phrase “over $k$” must remain visible in the definition. [example: Base Field Dependence] In $\mathbb{A}^1_{\mathbb{R}}$, a point is a real number $a\in \mathbb{R}$. For every real $a$, the square $a^2$ is nonnegative, so $a^2\geq 0$ and therefore \begin{align*} a^2+1\geq 1. \end{align*} Hence $a^2+1\neq 0$ for every $a\in \mathbb{R}$, and the real vanishing set is \begin{align*} V_{\mathbb{R}}(x^2+1)=\{a\in \mathbb{R}:a^2+1=0\}=\varnothing. \end{align*} In $\mathbb{A}^1_{\mathbb{C}}$, the same polynomial is evaluated on complex numbers. Since $i^2=-1$, we have \begin{align*} i^2+1=-1+1=0. \end{align*} Also $(-i)^2=(-1)^2i^2=i^2=-1$, so \begin{align*} (-i)^2+1=-1+1=0. \end{align*} Conversely, if $z\in \mathbb{C}$ satisfies $z^2+1=0$, then $z^2=-1=i^2$, so \begin{align*} z^2-i^2=0. \end{align*} Factoring the difference of squares gives \begin{align*} (z-i)(z+i)=0. \end{align*} Because $\mathbb{C}$ is a field, it has no zero divisors, so $z-i=0$ or $z+i=0$. Thus $z=i$ or $z=-i$, and hence \begin{align*} V_{\mathbb{C}}(x^2+1)=\{i,-i\}. \end{align*} The same formal equation therefore defines different affine varieties when the field of points changes. [/example] The final example shows why reducible varieties are included in the broad definition. A single equation may describe a union of simpler geometric pieces. [example: A Reducible Variety] Let $k$ be an infinite field and consider \begin{align*} X=V(xy) \subset \mathbb{A}^2_k. \end{align*} For a point $(a,b)\in \mathbb{A}^2_k$, the condition $(a,b)\in X$ means \begin{align*} xy(a,b)=ab=0. \end{align*} Since $k$ is a field, it has no zero divisors, so $ab=0$ holds exactly when $a=0$ or $b=0$. Hence \begin{align*} X=\{(0,b):b\in k\}\cup\{(a,0):a\in k\}=V(x)\cup V(y). \end{align*} Thus $X$ is the union of the two coordinate axes. We compute its vanishing ideal. Since $xy$ vanishes on every point of $X$, every multiple of $xy$ also vanishes on $X$, so \begin{align*} (xy)\subset I(X). \end{align*} Conversely, let $f\in I(X)$ and write \begin{align*} f=\sum_{i,j} c_{ij}x^i y^j \end{align*} with only finitely many nonzero coefficients $c_{ij}\in k$. On the $x$-axis we have $(a,0)\in X$ for every $a\in k$, so \begin{align*} 0=f(a,0)=\sum_i c_{i0}a^i \end{align*} for every $a\in k$. A nonzero one-variable polynomial over a field has only finitely many roots, while $k$ is infinite, so the polynomial $\sum_i c_{i0}x^i$ is zero and therefore $c_{i0}=0$ for every $i$. Similarly, on the $y$-axis we have $(0,b)\in X$ for every $b\in k$, so \begin{align*} 0=f(0,b)=\sum_j c_{0j}b^j \end{align*} for every $b\in k$. Again the infinite-field root bound gives $c_{0j}=0$ for every $j$. Therefore every nonzero monomial term of $f$ has both exponents positive, so \begin{align*} f=\sum_{i\geq 1,\ j\geq 1} c_{ij}x^i y^j=xy\sum_{i\geq 1,\ j\geq 1} c_{ij}x^{i-1}y^{j-1}. \end{align*} Thus $f\in(xy)$, and hence \begin{align*} I(X)=(xy). \end{align*} The coordinate ring is therefore \begin{align*} k[X]=k[x,y]/I(X)=k[x,y]/(xy). \end{align*} The relation $xy=0$ records that the two coordinate axes are separate away from the origin but meet where both $x=0$ and $y=0$. The infinite-field hypothesis is used in the vanishing-ideal computation: if $k=\mathbb{F}_q$, then every $a\in\mathbb{F}_q$ satisfies $a^q-a=0$, so $x^q-x$ and $y^q-y$ vanish on all $\mathbb{F}_q$-points of $X$, even though neither polynomial is a multiple of $xy$. [/example] ## Properties Affine varieties behave well under finite unions and arbitrary intersections. This is why they form the closed sets of the Zariski topology, which is the natural topology for algebraic geometry rather than the Euclidean topology. [quotetheorem:9519] The theorem says that algebraic geometry comes with its own topology. The closed sets are large compared with Euclidean closed sets; for instance, over an infinite field every nonempty open subset of $\mathbb{A}^1_k$ in the Zariski topology is the complement of a finite set. The next basic control problem is to understand how changing the equations changes the set of solutions. Adding equations can only remove points: a point that satisfies all equations in a larger ideal also satisfies the equations in any smaller ideal. This creates a reversal between algebraic containment and geometric containment, so one must keep careful track of the direction of inclusions when translating between ideals and zero sets. [quotetheorem:9520] The coordinate ring packages this reversal into algebra. To describe the maps that belong to algebraic geometry, we require coordinate functions to be polynomial rather than arbitrary set-theoretic functions. [definition: Morphism of Affine Varieties] Let $k$ be a field, let $X \subset \mathbb{A}^n_k$ and $Y \subset \mathbb{A}^m_k$ be affine varieties. A morphism of affine varieties from $X$ to $Y$ is a function $\phi: X \to Y$ for which there exist polynomials $f_1, \ldots, f_m \in k[x_1, \ldots, x_n]$ and a polynomial map $\Phi: \mathbb{A}^n_k \to \mathbb{A}^m_k$ with \begin{align*} \Phi(p) = (f_1(p), \ldots, f_m(p)) \end{align*} for every $p \in \mathbb{A}^n_k$, and such that $\Phi$ restricts to $\phi$ on $X$; that is, \begin{align*} \phi(p) = \Phi(p) \end{align*} for every $p \in X$. [/definition] A morphism is the algebraic-geometric substitute for a continuous map or smooth map, but the definition still appears to depend on chosen coordinate polynomials. The coordinate ring removes that ambiguity: polynomial functions on the target can be pulled back along a morphism to polynomial functions on the source, and this pullback reverses the direction of the map. This raises the precise algebraic question of how to recognize a geometric map using only coordinate rings. The formal correspondence below answers that question by turning morphisms of affine varieties into homomorphisms between their rings of polynomial functions. [quotetheorem:9521] This contravariant relationship is the prototype for the functorial viewpoint of modern algebraic geometry. It is also the reason affine schemes are built from rings: affine varieties already contain a ring-theoretic description of their maps. ## Relationship to Other Concepts Affine varieties are the affine part of algebraic geometry. Projective varieties add points at infinity and are better suited to compactness-like phenomena, intersection theory, and homogeneous equations. Many projective questions are studied by covering a [projective variety](/page/Projective%20Variety) with affine charts, each of which is controlled by an affine coordinate ring. The Zariski Topology is defined so that affine varieties are closed sets in affine space. This topology is coarser than the usual topology on $\mathbb{R}^n$ or $\mathbb{C}^n$, but it is better adapted to polynomial equations: polynomial functions are continuous for the Zariski topology, and irreducibility becomes a topological property. The Coordinate Ring of an affine variety is the algebraic object that stores its polynomial functions. For algebraically closed fields, the affine variety can often be recovered from this ring up to isomorphism, and morphisms of varieties correspond to homomorphisms of coordinate rings in the opposite direction. The [Ideal](/page/Ideal) attached to equations determines the zero set, but the exact ideal may contain nilpotent information that the point set cannot see. Passing from $I$ to its radical removes this invisible information. Schemes restore and organize that information, which is why Scheme theory extends the classical theory of affine varieties. Affine varieties also connect to Commutative Algebra. Dimension, irreducible components, singular points, and maps can all be studied through ideals, prime ideals, local rings, and modules. This is not a side connection: it is the main computational engine of the subject. ## Common Pitfalls A first pitfall is to treat the defining equations as part of the variety. The affine variety is the set $V(S)$, not the chosen list $S$. The equations $y-x^2=0$ and $y^3-x^2y^2=0$ may have related zero sets in some contexts but encode different ideals; the vanishing ideal is the canonical algebraic object attached to the point set. A second pitfall is to forget the base field. The equation $x^2+1=0$ cuts out no real points in $\mathbb{A}^1_{\mathbb{R}}$ but two complex points in $\mathbb{A}^1_{\mathbb{C}}$. Algebraic geometry handles this by carefully tracking the field of definition, field extensions, and rational points. A third pitfall is to assume that all affine varieties are irreducible. The variety $V(xy)$ is a union of two axes, and reducible examples occur naturally. Irreducible affine varieties are important enough to deserve their own definition, but the broader class is needed for unions, intersections, and Zariski closed sets. ## Beyond and Connections Affine varieties are the entry point to the dictionary between algebra and geometry: polynomial ideals encode geometric loci, while geometric operations such as union, intersection, and projection have algebraic shadows in operations on ideals and coordinate rings. This viewpoint is the foundation for later refinements such as irreducible varieties, morphisms of varieties, dimension, singularities, and projective varieties. The affine case is deliberately concrete, but it already contains the central idea of algebraic geometry: studying spaces by studying the functions and equations that define them. ## References Atiyah and Macdonald, *Introduction to Commutative Algebra* (1969). Cox, Little, and O'Shea, *Ideals, Varieties, and Algorithms* (2015). Hartshorne, *Algebraic Geometry* (1977). Shafarevich, *Basic Algebraic Geometry 1* (2013).