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Krull's Theorem
In mathematics, and more specifically in ring theory, Krull's theorem, named after Wolfgang Krull, asserts that a nonzero ring has at least one maximal ideal. The theorem was proved in 1929 by Krull, who used transfinite induction. The theorem admits a simple proof using Zorn's lemma, and in fact is equivalent to Zorn's lemma, which in turn is equivalent to the axiom of choice. Variants * For noncommutative rings, the analogues for maximal left ideals and maximal right ideals also hold. * For pseudo-rings, the theorem holds for regular ideals. * A slightly stronger (but equivalent) result, which can be proved in a similar fashion, is as follows: :::Let ''R'' be a ring, and let ''I'' be a proper ideal of ''R''. Then there is a maximal ideal of ''R'' containing ''I''. :This result implies the original theorem, by taking ''I'' to be the zero ideal (0). Conversely, applying the original theorem to ''R''/''I'' leads to this result. :To prove the stronger result directly, consi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudo-ring
In mathematics, and more specifically in abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ..., a pseudo-ring is one of the following variants of a ring (mathematics), ring: * A rng (algebra), rng, i.e., a structure satisfying all the axioms of a ring except for the existence of a multiplicative identity. * A set ''R'' with two binary operations + and ⋅ such that is an abelian group with additive identity, identity 0, and and for all ''a'', ''b'', ''c'' in ''R''. * An abelian group equipped with a subgroup ''B'' and a multiplication making ''B'' a ring and ''A'' a ''B''-module (mathematics), module. None of these definitions are equivalent, so it is best to avoid the term "pseudo-ring" or to clarify which meaning is intended. See also * Semiring – an algebra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
