Prime idealIn algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with the zero ideal. Primitive ideals are prime, and prime ideals are both primary and semiprime. An ideal P of a commutative ring R is prime if it has the following two properties: If a and b are two elements of R such that their product ab is an element of P, then a is in P or b is in P, P is not the whole ring R.
Ideal (ring theory)In mathematics, and more specifically in ring theory, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any integer (even or odd) results in an even number; these closure and absorption properties are the defining properties of an ideal.
Maximal idealIn mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all proper ideals. In other words, I is a maximal ideal of a ring R if there are no other ideals contained between I and R. Maximal ideals are important because the quotients of rings by maximal ideals are simple rings, and in the special case of unital commutative rings they are also fields.
Radical of an idealIn ring theory, a branch of mathematics, the radical of an ideal of a commutative ring is another ideal defined by the property that an element is in the radical if and only if some power of is in . Taking the radical of an ideal is called radicalization. A radical ideal (or semiprime ideal) is an ideal that is equal to its radical. The radical of a primary ideal is a prime ideal. This concept is generalized to non-commutative rings in the Semiprime ring article.
Primary decompositionIn mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be decomposed as an intersection, called primary decomposition, of finitely many primary ideals (which are related to, but not quite the same as, powers of prime ideals). The theorem was first proven by for the special case of polynomial rings and convergent power series rings, and was proven in its full generality by .
Minimal prime idealIn mathematics, especially in commutative algebra, certain prime ideals called minimal prime ideals play an important role in understanding rings and modules. The notion of height and Krull's principal ideal theorem use minimal primes. A prime ideal P is said to be a minimal prime ideal over an ideal I if it is minimal among all prime ideals containing I. (Note: if I is a prime ideal, then I is the only minimal prime over it.) A prime ideal is said to be a minimal prime ideal if it is a minimal prime ideal over the zero ideal.
Geometric invariant theoryIn mathematics, geometric invariant theory (or GIT) is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces. It was developed by David Mumford in 1965, using ideas from the paper in classical invariant theory. Geometric invariant theory studies an action of a group G on an algebraic variety (or scheme) X and provides techniques for forming the 'quotient' of X by G as a scheme with reasonable properties.
Knot polynomialIn the mathematical field of knot theory, a knot polynomial is a knot invariant in the form of a polynomial whose coefficients encode some of the properties of a given knot. The first knot polynomial, the Alexander polynomial, was introduced by James Waddell Alexander II in 1923. Other knot polynomials were not found until almost 60 years later. In the 1960s, John Conway came up with a skein relation for a version of the Alexander polynomial, usually referred to as the Alexander–Conway polynomial.
Ideal theoryIn mathematics, ideal theory is the theory of ideals in commutative rings. While the notion of an ideal exists also for non-commutative rings, a much more substantial theory exists only for commutative rings (and this article therefore only considers ideals in commutative rings.) Throughout the articles, rings refer to commutative rings. See also the article ideal (ring theory) for basic operations such as sum or products of ideals.
Fractional idealIn mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral domain are like ideals where denominators are allowed. In contexts where fractional ideals and ordinary ring ideals are both under discussion, the latter are sometimes termed integral ideals for clarity. Let be an integral domain, and let be its field of fractions.
Associated primeIn abstract algebra, an associated prime of a module M over a ring R is a type of prime ideal of R that arises as an annihilator of a (prime) submodule of M. The set of associated primes is usually denoted by and sometimes called the assassin or assassinator of M (word play between the notation and the fact that an associated prime is an annihilator). In commutative algebra, associated primes are linked to the Lasker–Noether primary decomposition of ideals in commutative Noetherian rings.
Alexander polynomialIn mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomial, in 1923. In 1969, John Conway showed a version of this polynomial, now called the Alexander–Conway polynomial, could be computed using a skein relation, although its significance was not realized until the discovery of the Jones polynomial in 1984.
Jones polynomialIn the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynomial in the variable with integer coefficients. Suppose we have an oriented link , given as a knot diagram. We will define the Jones polynomial, , using Louis Kauffman's bracket polynomial, which we denote by . Here the bracket polynomial is a Laurent polynomial in the variable with integer coefficients.
Splitting of prime ideals in Galois extensionsIn mathematics, the interplay between the Galois group G of a Galois extension L of a number field K, and the way the prime ideals P of the ring of integers OK factorise as products of prime ideals of OL, provides one of the richest parts of algebraic number theory. The splitting of prime ideals in Galois extensions is sometimes attributed to David Hilbert by calling it Hilbert theory. There is a geometric analogue, for ramified coverings of Riemann surfaces, which is simpler in that only one kind of subgroup of G need be considered, rather than two.
Knot invariantIn the mathematical field of knot theory, a knot invariant is a quantity (in a broad sense) defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some invariants are indeed numbers (algebraic), but invariants can range from the simple, such as a yes/no answer, to those as complex as a homology theory (for example, "a knot invariant is a rule that assigns to any knot K a quantity φ(K) such that if K and are equivalent then φ(K) = φ().
Principal idealIn mathematics, specifically ring theory, a principal ideal is an ideal in a ring that is generated by a single element of through multiplication by every element of The term also has another, similar meaning in order theory, where it refers to an (order) ideal in a poset generated by a single element which is to say the set of all elements less than or equal to in The remainder of this article addresses the ring-theoretic concept.
HOMFLY polynomialIn the mathematical field of knot theory, the HOMFLY polynomial or HOMFLYPT polynomial, sometimes called the generalized Jones polynomial, is a 2-variable knot polynomial, i.e. a knot invariant in the form of a polynomial of variables m and l. A central question in the mathematical theory of knots is whether two knot diagrams represent the same knot. One tool used to answer such questions is a knot polynomial, which is computed from a diagram of the knot and can be shown to be an invariant of the knot, i.e.
Quantum invariantIn the mathematical field of knot theory, a quantum knot invariant or quantum invariant of a knot or link is a linear sum of colored Jones polynomial of surgery presentations of the knot complement. Finite type invariant Kontsevich invariant Kashaev's invariant Witten–Reshetikhin–Turaev invariant (Chern–Simons) Invariant differential operator Rozansky–Witten invariant Vassiliev knot invariant Dehn invariant LMO invariant Turaev–Viro invariant Dijkgraaf–Witten invariant Reshetikhin–Turaev invariant Tau-inva
Structure theorem for finitely generated modules over a principal ideal domainIn mathematics, in the field of abstract algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that finitely generated modules over a principal ideal domain (PID) can be uniquely decomposed in much the same way that integers have a prime factorization. The result provides a simple framework to understand various canonical form results for square matrices over fields.
Hilbert's basis theoremIn mathematics, specifically commutative algebra, Hilbert's basis theorem says that a polynomial ring over a Noetherian ring is Noetherian. If is a ring, let denote the ring of polynomials in the indeterminate over . Hilbert proved that if is "not too large", in the sense that if is Noetherian, the same must be true for . Formally, Hilbert's Basis Theorem. If is a Noetherian ring, then is a Noetherian ring. Corollary. If is a Noetherian ring, then is a Noetherian ring.
