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Algebraic Proof Terms

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

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Variety

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

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Scheme

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

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Morphism

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

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Rational Map

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A map between varieties defined by ratios of polynomial functions.

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Ring

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

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Field

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

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Module

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

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Tangent Space

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

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Homomorphism

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A map between two algebraic structures of the same type that respects the structure.

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