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In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory equivalent to the statement that a Cartesian product of a collection of non-empty sets is non-empty. Informally put, the axiom of choice says that given any collection of sets, each containing at least one element, it is possible to construct a new set by choosing one element from each set, even if the collection is infinite. Formally, it states that for every indexed family of nonempty sets, there exists an indexed set such that for every . The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem.
In 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.
In mathematics, specifically in ring theory, the simple modules over a ring R are the modules over R that are non-zero and have no non-zero proper submodules. Equivalently, a module M is simple if and only if every cyclic submodule generated by a non-zero element of M equals M. Simple modules form building blocks for the modules of finite length, and they are analogous to the simple groups in group theory.
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal 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.
Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain necessarily contains at least one maximal element.
In commutative algebra, the Krull dimension of a commutative ring R, named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally the Krull dimension can be defined for modules over possibly non-commutative rings as the deviation of the poset of submodules.
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noetherian respectively. That is, every increasing sequence of left ideals has a largest element; that is, there exists an n such that:
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings.
In mathematics, more specifically in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on algebraic varieties or manifolds, or of algebraic number fields examined at a particular place, or prime. Local algebra is the branch of commutative algebra that studies commutative local rings and their modules.
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings; rings of algebraic integers, including the ordinary integers ; and p-adic integers.
In algebra, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings; their representations, or, in different language, modules; special classes of rings ; related structures like rngs; as well as an array of properties that prove to be of interest both within the theory itself and for its applications, such as homological properties and polynomial identities.
In mathematics, a transcendental extension is a field extension such that there exists an element in the field that is transcendental over the field ; that is, an element that is not a root of any univariate polynomial with coefficients in . In other words, a transcendental extension is a field extension that is not algebraic. For example, and are both transcendental extensions of
In mathematics, the Boolean prime ideal theorem states that ideals in a Boolean algebra can be extended to prime ideals. A variation of this statement for filters on sets is known as the ultrafilter lemma. Other theorems are obtained by considering different mathematical structures with appropriate notions of ideals, for example, rings and prime ideals, or distributive lattices and maximal ideals. This article focuses on prime ideal theorems from order theory.
In commutative algebra, Krull's principal ideal theorem, named after Wolfgang Krull (1899–1971), gives a bound on the height of a principal ideal in a commutative Noetherian ring. The theorem is sometimes referred to by its German name, Krulls Hauptidealsatz.
In abstract algebra, a valuation ring is an integral domain D such that for every non-zero element x of its field of fractions F, at least one of x or x−1 belongs to D.
In mathematics, more specifically abstract algebra and commutative algebra, Nakayama's lemma — also known as the Krull–Azumaya theorem — governs the interaction between the Jacobson radical of a ring and its finitely generated modules. Informally, the lemma immediately gives a precise sense in which finitely generated modules over a commutative ring behave like vector spaces over a field. It is an important tool in algebraic geometry, because it allows local data on algebraic varieties, in the form of modules over local rings, to be studied pointwise as vector spaces over the residue field of the ring.
In 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 prime ideals.
In mathematics, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exist a and b in the ring such that ab and ba are different. Equivalently, a noncommutative ring is a ring that is not a commutative ring.
In algebra, Zariski's lemma, proved by Oscar Zariski, states that, if a field K is finitely generated as an associative algebra over another field k, then K is a finite field extension of k.
In the mathematical field of set theory, an ultrafilter on a set is a maximal filter on the set In other words, it is a collection of subsets of that satisfies the definition of a filter on and that is maximal with respect to inclusion, in the sense that there does not exist a strictly larger collection of subsets of that is also a filter. Equivalently, an ultrafilter on the set can also be characterized as a filter on with the property that for every subset of either or its complement belongs to the ultrafilter.
References
- Krull, W. (1929). "Idealtheorie in Ringen ohne Endlichkeitsbedingungen". Mathematische Annalen . 101 (1): 729–744. doi:10.1007/BF01454872. S2CID 119883473.
- Hodges, W. (1979). "Krull implies Zorn". Journal of the London Mathematical Society . s2-19 (2): 285–287. doi:10.1112/jlms/s2-19.2.285.