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A relation "<=" is called a preorder (or quasiorder) on a set S if it satisfies: 1. Reflexivity: a<=a for all a in S. 2. Transitivity: a<=b and b<=c implies a<=c. A preorder ...

A primary ideal is an ideal I such that if ab in I, then either a in I or b^m in I for some m>0. Prime ideals are always primary. A primary decomposition expresses any ideal ...

The primitive part of a polynomial P(x) is P(x)/k, where k is the content. For a general univariate polynomial P(x), the Wolfram Language function FactorTermsList[poly, x] ...

The derivative identity d/(dx)[f(x)g(x)] = lim_(h->0)(f(x+h)g(x+h)-f(x)g(x))/h (1) = (2) = lim_(h->0)[f(x+h)(g(x+h)-g(x))/h+g(x)(f(x+h)-f(x))/h] (3) = f(x)g^'(x)+g(x)f^'(x), ...

Given a vector space V, its projectivization P(V), sometimes written P(V-0), is the set of equivalence classes x∼lambdax for any lambda!=0 in V-0. For example, complex ...

An extension A subset B of a group, ring, module, field, etc., such that A!=B.

A submodule that is not the entire module.

A subvariety of an algebraic variety that is not the entire variety. The proper subvarieties of a line are its nonempty finite subsets, the proper subvarieties of a plane are ...

A proposition is a mathematical statement such as "3 is greater than 4," "an infinite set exists," or "7 is prime." An axiom is a proposition that is assumed to be true. With ...

A function f defined on a subset S subset R^n is said to be pseudoconcave if -f is pseudoconvex.