hyperreal ideal

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• Hyperreal number — *R redirects here. For R*, see Rockstar Games. The system of hyperreal numbers represents a rigorous method of treating the infinite and infinitesimal quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R… …   Wikipedia

• Valuation ring — In abstract algebra, a valuation ring is an integral domain D such that for every element x of its field of fractions F , at least one of x or x 1 belongs to D .Given a field F , if D is a subring of F such that either x or x 1 belongs to D for… …   Wikipedia

• Non-standard analysis — Abraham Robinson Gottfried Wilhelm Leibniz argued tha …   Wikipedia

• Quotient ring — In mathematics a quotient ring, also known as factor ring or residue class ring, is a construction in ring theory, quite similar to the factor groups of group theory and the quotient spaces of linear algebra. One starts with a ring R and a two… …   Wikipedia

• Number — For other uses, see Numbers (disambiguation). A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational… …   Wikipedia

• Ultrafilter — In the mathematical field of set theory, an ultrafilter on a set X is a collection of subsets of X that is a filter, that cannot be enlarged (as a filter). An ultrafilter may be considered as a finitely additive measure. Then every subset of X is …   Wikipedia

• Dual number — For dual grammatical number found in some languages, see Dual (grammatical number). In linear algebra, the dual numbers extend the real numbers by adjoining one new element ε with the property ε2 = 0 (ε is nilpotent). The collection of dual… …   Wikipedia

• Construction of the real numbers — In mathematics, there are several ways of defining the real number system as an ordered field. The synthetic approach gives a list of axioms for the real numbers as a complete ordered field. Under the usual axioms of set theory, one can show that …   Wikipedia

• Differential (infinitesimal) — For other uses of differential in calculus, see differential (calculus), and for more general meanings, see differential. In calculus, a differential is traditionally an infinitesimally small change in a variable. For example, if x is a variable …   Wikipedia

• Superreal number — The superreal numbers are an extension of the real numbers, similar to the surreal numbers or hyperreal numbers, but comprising a more inclusive category than either one.Formal DefinitionSuppose X is a Tychonoff space, also called a T3.5 space,… …   Wikipedia

• Real closed field — In mathematics, a real closed field is a field F in which any of the following equivalent conditions are true:#There is a total order on F making it an ordered field such that, in this ordering, every positive element of F is a square in F and… …   Wikipedia