Universal algebraic geometry
Encyclopedia
In universal algebraic geometry, algebraic geometry
is generalized from the geometry of rings
to geometry of arbitrary varieties of algebras
, so that every variety of algebra has its own algebraic geometry. Note that the two terms algebraic variety
and variety of algebra should not be confused.
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
is generalized from the geometry of rings
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
to geometry of arbitrary varieties of algebras
Variety (universal algebra)
In mathematics, specifically universal algebra, a variety of algebras is the class of all algebraic structures of a given signature satisfying a given set of identities. Equivalently, a variety is a class of algebraic structures of the same signature which is closed under the taking of homomorphic...
, so that every variety of algebra has its own algebraic geometry. Note that the two terms algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
and variety of algebra should not be confused.