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A geometric object defined as the set of solutions to a system of polynomial equations.

A map between varieties defined by ratios of polynomial functions.

A set of polynomials within a ring that is closed under addition and under multiplication by any polynomial in the ring.

An advanced mathematical structure that enlarges the notion of algebraic variety to include multiplicities and 'bad' intersection points.

A generalization of the concept of vector space over a field, where the field is replaced with a ring.

A structure-preserving map between two algebraic objects, like varieties or schemes.

An algebraic structure consisting of a set equipped with two binary operations that generalize the arithmetic operations of addition and multiplication.

A commutative ring in which every non-zero element has a multiplicative inverse.

At a point on a differentiable manifold, it is a vector space that consists of the tangent vectors to the manifold at that point.