Irreducible components of affine varieties are the basic building blocks of an affine variety. They are the irreducible algebraic sets that cannot be further decomposed into smaller algebraic sets.
Irreducible components of affine varieties correspond to the minimal prime ideals of the variety's coordinate ring. Each irreducible component corresponds to a unique minimal prime ideal.
Yes, an affine variety can have multiple irreducible components. In fact, most affine varieties have more than one irreducible component. However, some special varieties, such as smooth curves, have only one irreducible component.
The dimension of an affine variety is equal to the maximum dimension of its irreducible components. This means that the dimension of an affine variety is determined by the dimension of its largest irreducible component.
The Zariski closure of an affine variety is the smallest algebraic set that contains the variety. The irreducible components of the affine variety are exactly the irreducible components of its Zariski closure. This means that the Zariski closure can be seen as the union of all irreducible components of the affine variety.