ideals - Article Index

Dictionary - Conflict Dream Symbol: Dreams Interpretation Dictionary - Conflict
Conflict - Fighting   (1) A fight may represent conflict in a real-life domestic or work situation.   (2) The conflict may b...   » Read the article

Dictionary - Dancing Dream Symbol: Dreams Interpretation Dictionary - Dancing
Dancing (1) By virtue of its sexual associations, dancing may represent a coming together - courtship - of the masculine (animus) and ...   » Read the article

Encyclopedia - Complete Lattice: Encyclopedia - Complete Lattice
In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). Com...   » Read the article

Encyclopedia - Coprime: Encyclopedia - Coprime
In mathematics, the integers a and b are said to be coprime or relatively prime if they have no common factor other than 1 and −1, or e...   » Read the article

Encyclopedia - Christian Left: Encyclopedia - Christian Left
The Christian Left encompasses those who hold a strong Christian belief and share left-wing or socialist ideals. Contrary to the situatio...   » Read the article

Encyclopedia - Idealism: Encyclopedia - Idealism
Idealism is an approach to philosophical enquiry. The ideal, in these systems, relates to direct knowledge of subjective mental ideas, or...   » Read the article

Encyclopedia - Zorn's Lemma: Encyclopedia - Zorn's Lemma
Zorn's lemma, also known as the Kuratowski-Zorn lemma, is a theorem of set theory that states: Every non-empty partially ordered set in w...   » Read the article

Encyclopedia - Congruence Relation: Encyclopedia - Congruence Relation
In mathematics and especially in abstract algebra, a congruence relation or simply congruence is an equivalence relation that is compatib...   » Read the article

Encyclopedia - Artinian Ring: Encyclopedia - Artinian Ring
In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals. There are two classes of rings t...   » Read the article

Encyclopedia - Body Image: Encyclopedia - Body Image
Body image is a person's perception of his or her physical appearance. A person with a poor body image will perceive his or her own body ...   » Read the article

Encyclopedia - Meaning Of Life: Encyclopedia - Meaning Of Life
The philosophical question "What is the meaning of life?" means different things to different people. The vagueness of the query is inher...   » Read the article

Encyclopedia - Idealism: Encyclopedia Ii - Idealism - History
Idealism names a number of philosophical positions with quite different tendencies and implications. Idealism - Plato. Plato proposed a...   » Read the article

Encyclopedia - Chinese Remainder Theorem: Encyclopedia Ii - Chinese Remainder Theorem - Simultaneous Congruences Of Integers
The original form of the theorem, contained in a third-century book by Chinese mathematician Sun Tzu and later republished in a 1247 book...   » Read the article

Encyclopedia - Boolean Prime Ideal Theorem: Encyclopedia Ii - Boolean Prime Ideal Theorem - Boolean Prime Ideal Theorem
The Boolean prime ideal theorem is the strong prime ideal theorem for Boolean algebras. Thus the formal statement is: Let B be a Boolean...   » Read the article

Encyclopedia - Heian Period: Encyclopedia Ii - Heian Period - The Fujiwara Regency
When Kammu moved the capital to Heian (Kyoto), which remained the imperial capital for the next 1,000 years, he did so not only to streng...   » Read the article

Encyclopedia - Filter Mathematics: Encyclopedia Ii - Filter Mathematics - Filter On A Set
A special case of a filter is a filter defined on a set. Given a set S, a partial ordering ⊆ can be defined on the powerset P(S) by sub...   » Read the article

Encyclopedia - Ideology: Encyclopedia Ii - Ideology - The Analysis Of Ideology
Meta-ideology is the study of the structure, form, and manifestation of ideologies. Meta-ideology posits that ideology is a coherant syst...   » Read the article

Encyclopedia - Essentialism: Encyclopedia Ii - Essentialism - Essentialism In Philosophy
The definition, in philosophical contexts, of the word "essence" is very close to the definition of form (Gr. eidos). Many definitions of...   » Read the article

Encyclopedia - Equivalence Of Categories: Encyclopedia Ii - Equivalence Of Categories - Equivalent Characterizations
One can show that a functor F : C -> D yields an equivalence of categories if and only if has all of the following three properti...   » Read the article

Encyclopedia - Body Image: Encyclopedia Ii - Body Image - Research: Measuring Body Image
Body image is often measured by asking the subject to rate his or her current and ideal body shape using a series of drawings of increasi...   » Read the article

Encyclopedia - Ring Theory: Encyclopedia Ii - Ring Theory - Elementary Introduction
Ring theory - Definition. Formally, a ring is an abelian group (R, +), together with a second binary operation * such that for all a, b...   » Read the article

Encyclopedia - Realism In International Relations: Encyclopedia Ii - Realism In International Relations - Basic Theory
Fundamental principles common to realist theories: The international system is anarchical. Sovereign states are the principal actors in ...   » Read the article

Encyclopedia - Sieve Category Theory: Encyclopedia Ii - Sieve Category Theory - Pullback Of Sieves
The most common operation on a sieve is pullback. Pulling back a sieve S on c by an arrow f:c′→c gives a new sieve f*S on c′. This ...   » Read the article

Encyclopedia - Zorn's Lemma: Encyclopedia Ii - Zorn's Lemma - An Example Application
We will go over a typical application of Zorn's lemma: the proof that every ring R contains a maximal ideal. The set P here consists of a...   » Read the article

Encyclopedia - Isomorphism Theorem: Encyclopedia Ii - Isomorphism Theorem - Groups
First we state the isomorphism theorems for groups, where they take a simpler form and state important properties of quotient groups (als...   » Read the article

Encyclopedia - Polynomial: Encyclopedia Ii - Polynomial - Elementary Properties Of Polynomials
All polynomials have an expanded form, in which the distributive law has been used to remove all parentheses. (Some polynomials also have...   » Read the article

Encyclopedia - Meaning Of Life: Encyclopedia Ii - Meaning Of Life - Philosophical Views
Meaning of life - Value as meaning. In that they attempt to answer the question "What is valuable in life?", theories of value are theo...   » Read the article

Encyclopedia - Order Theory: Encyclopedia Ii - Order Theory - Introduction To The Basic Definitions
This section aims at giving a first guide to the realm of ordered sets. It addresses readers who have basic knowledge of set theory and a...   » Read the article

Encyclopedia - Preadditive Category: Encyclopedia Ii - Preadditive Category - Elementary Properties
Because every hom-set Hom(A,B) is an abelian group, it has a zero element 0. This is the zero morphism from A to B. Because composition o...   » Read the article

Encyclopedia - Field Of Sets: Encyclopedia Ii - Field Of Sets - Fields Of Sets With Additional Structure
Field of sets - Sigma algebras and measure spaces. If an algebra over a set is closed under countable intersections and countable union...   » Read the article

Encyclopedia - Field Mathematics: Encyclopedia Ii - Field Mathematics - Definition
A field is a commutative ring (F, +, *) such that 0 does not equal 1 and all elements of F except 0 have a multiplicative inverse. Spelle...   » Read the article

Encyclopedia - Christian Left: Encyclopedia Ii - Christian Left - History
Christian left - Early Christianity as anti-establishment movement. See also: Christian anarchism Early Christian anti-materialism O...   » Read the article

Encyclopedia - Body Image: Encyclopedia Ii - Body Image - Information On Specific Minority Populations
Most empirical research and statistical data are oriented and tailored toward Caucasian audiences, and some studies have been designed to...   » Read the article

Encyclopedia - Ideology: Encyclopedia Ii - Ideology - The Analysis Of Ideology
Meta-ideology is the study of the structure, form, and manifestation of ideologies. Meta-ideology posits that ideology is a coherent syst...   » Read the article

Encyclopedia - Congruence Relation: Encyclopedia Ii - Congruence Relation - Group Theory
In the particular case of groups, congruence relations can be described in elementary terms as follows: If G is a group (with identity el...   » Read the article

Encyclopedia - Complete Lattice: Encyclopedia Ii - Complete Lattice - Free Construction And Completion
Complete lattice - Free complete semilattices. As usual, the construction of free objects depends on the chosen class of morphisms. Let...   » Read the article

Encyclopedia - Idealism: Encyclopedia Ii - Idealism - History
Idealism names a number of philosophical positions with quite different tendencies and implications. Idealism - Idealism in the East. S...   » Read the article

Encyclopedia - Polynomial Ring: Encyclopedia Ii - Polynomial Ring - The Polynomial Ring In Several Variables
Given two variables X and Y, one constructs the polynomial ring R[X], and then, on top of it, the ring (R[X])[Y]. This ring is considered...   » Read the article

Encyclopedia - Polynomial: Encyclopedia Ii - Polynomial - Elementary Properties Of Polynomials
All polynomials have an expanded form, in which the distributive law has been used to remove all parentheses. (Some polynomials also have...   » Read the article

Encyclopedia - Emanuel Lasker: Encyclopedia Ii - Emanuel Lasker - Chess Champion
In 1894 he became the second World Chess Champion by defeating Steinitz with 10 wins, 4 draws and 5 losses. He maintained this title for ...   » Read the article

Encyclopedia - Decadence: Encyclopedia Ii - Decadence - Leninist Use
Vladimir Lenin continued and extended the use of the word "decadence" in his theory of imperialism to refer to economic matters underlyin...   » Read the article

Encyclopedia - Ideal Number: Encyclopedia Ii - Ideal Number - Example
For instance, let y be a root of y2 + y + 6 = 0, then the ring of integers of the field is , which means all a + by with a and b integer...   » Read the article

Encyclopedia - Coprime: Encyclopedia Ii - Coprime - Properties
There are a number of conditions which are equivalent to a and b being coprime: There exist integers x and y such that ax + by = 1 (see ...   » Read the article

Encyclopedia - Noetherian Ring: Encyclopedia Ii - Noetherian Ring - Introduction
Rings of polynomials over fields have many special properties; properties that follow from the fact that polynomial rings are not, in som...   » Read the article

Encyclopedia - Semiring: Encyclopedia Ii - Semiring - Definition
A semiring is a set R equipped with two binary operations + and ·, called addition and multiplication, such that: (R, +) is a commutati...   » Read the article

Encyclopedia - Complete Lattice: Encyclopedia Ii - Complete Lattice - Formal Definition
A partially ordered set (L, ≤) is a complete lattice if every subset A of L has both a greatest lower bound (infimum, meet) and a least...   » Read the article

Encyclopedia - Polynomial Ring: Encyclopedia Ii - Polynomial Ring - The Polynomial Ring R[x]
One can then check that the set of all polynomials with coefficients in the ring R, together with the addition + and the multiplication ...   » Read the article

Encyclopedia - Boolean Prime Ideal Theorem: Encyclopedia Ii - Boolean Prime Ideal Theorem - Applications
Intuitively, the Boolean prime ideal theorem states that there are "enough" prime ideals in a Boolean algebra in the sense that we can ex...   » Read the article

Encyclopedia - Sieve Category Theory: Encyclopedia Ii - Sieve Category Theory - Definition
Let C be a category, and let c be an object of C. A sieve S on c is a subfunctor of Hom(−, c), i.e., for all objects c′ of C, S(c′)...   » Read the article

Encyclopedia - Polynomial Ring: Encyclopedia Ii - Polynomial Ring - Definition Of A Polynomial
In real analysis, a polynomial is a certain type of a function of one or several variables (see polynomial), or in other words, a polynom...   » Read the article

Encyclopedia - Realism In International Relations: Encyclopedia Ii - Realism In International Relations - History Of Realism
Sun Tzu Thucydides The ancient Greek historian Thucydides, who wrote the History of the Peloponnesian War, is also cited as an intellectu...   » Read the article

Encyclopedia - Ideology: Encyclopedia Ii - Ideology - Political Ideologies
In social studies, a political ideology is a certain ethical, set of ideals, principles, doctrines, myths or symbols of a social movement...   » Read the article

Encyclopedia - Body Image: Encyclopedia Ii - Body Image - Causes And Influences
As clearly evidenced by literature around the world, poor body images have existed at least since the widespread availability of mirrors,...   » Read the article

Encyclopedia - Polynomial: Encyclopedia Ii - Polynomial - Graphs
A polynomial function in one real variable can be represented by a graph. The graph of the zero polynomial f(x) = 0 is the x-axis. T...   » Read the article

Encyclopedia - Noetherian Ring: Encyclopedia Ii - Noetherian Ring - Uses Of Noetherian Rings
The Noetherian property is central in ring theory and in areas that make heavy use of rings, such as algebraic geometry. The reason behin...   » Read the article

Encyclopedia - Meaning Of Life: Encyclopedia Ii - Meaning Of Life - Popular Beliefs
"What is the meaning of life?" is a question many people ask themselves at some point during their lives. Some people believe that the me...   » Read the article

Encyclopedia - Noetherian Ring: Encyclopedia Ii - Noetherian Ring - Characterizations Of Noetherian Rings
There are other, equivalent, definitions for a ring R to be left-Noetherian: Every left ideal I in R is finitely generated, i.e. there e...   » Read the article

Encyclopedia - Body Image: Encyclopedia Ii - Body Image - Attractiveness And Social Issues
Some researchers also found that men judge the female figure they found most attractive as heavier than women's ratings of the ideal body...   » Read the article

Encyclopedia - Body Image: Encyclopedia Ii - Body Image - Relationship To Psychological Disorders
Poor body images can often contribute to the onset of a variety of eating disorders, including anorexia nervosa, bulimia, and binge eatin...   » Read the article

Encyclopedia - Realism In International Relations: Encyclopedia Ii - Realism In International Relations - Modern Realism
Realism in international relations - Classical realism. George F. Kennan - Containment Nicholas Spykman - Geostrategy, Containment Her...   » Read the article

Encyclopedia - Ring Theory: Encyclopedia Ii - Ring Theory - History
The study of rings originated from the theory of polynomial rings and the theory of algebraic integers. Furthermore, the appearance of hy...   » Read the article

Encyclopedia - Isomorphism Theorem: Encyclopedia Ii - Isomorphism Theorem - Rings And Modules
The isomorphism theorems are also valid for modules over a fixed ring R (and therefore also for vector spaces over a fixed field). One ha...   » Read the article

Encyclopedia - Preadditive Category: Encyclopedia Ii - Preadditive Category - Kernels And Cokernels
Because the hom-sets in a preadditive category have zero morphisms, the notion of kernel and cokernel make sense. That is, if f: A&#...   » Read the article

Encyclopedia - Field Of Sets: Encyclopedia Ii - Field Of Sets - Fields Of Sets In The Representation Theory Of Boolean Algebras
Field of sets - Stone representation. Every finite Boolean algebra can be represented as a whole power set - the power set of its set o...   » Read the article

Encyclopedia - Ideology: Encyclopedia Ii - Ideology - Ideology In Everyday Society
In public discussions, some ideas seem to arise more commonly than others. Indeed, often completely separate people may be found to think...   » Read the article

Encyclopedia - Semiring: Encyclopedia Ii - Semiring - Semiring Theory
Much of the theory of rings continues to make sense when applied to arbitrary semirings. In particular, one can generalise the theory of ...   » Read the article

Encyclopedia - Idealism: Encyclopedia Ii - Idealism - Other Uses
In general parlance, "idealism" or "idealist" is also used to describe a person having high ideals, sometimes with the connotation that t...   » Read the article

Encyclopedia - Preadditive Category: Encyclopedia Ii - Preadditive Category - Biproducts
Any finite product in a preadditive category must also be a coproduct, and conversely. In fact, finite products and coproducts in additiv...   » Read the article

Encyclopedia - Preadditive Category: Encyclopedia Ii - Preadditive Category - Additive Functors
If C and D are preadditive categories, then a functor F: C → D is additive if it too is enriched over the category Ab. T...   » Read the article

Encyclopedia - Chinese Remainder Theorem: Encyclopedia Ii - Chinese Remainder Theorem - Statement For General Rings
One of the most general forms of the Chinese remainder theorem can be formulated for rings and (two-sided) ideals. If R is a ring and I1,...   » Read the article

Encyclopedia - Boolean Prime Ideal Theorem: Encyclopedia Ii - Boolean Prime Ideal Theorem - Further Prime Ideal Theorems
The prototypical properties that were discussed for Boolean algebras in the above section can easily be modified to include more general ...   » Read the article

Encyclopedia - Chinese Remainder Theorem: Encyclopedia Ii - Chinese Remainder Theorem - Statement For Principal Ideal Domains
For a principal ideal domain R the Chinese remainder theorem takes the following form: If u1, ..., uk are elements of R which are pairwis...   » Read the article

Encyclopedia - Preadditive Category: Encyclopedia Ii - Preadditive Category - Examples
The most obvious example of a preadditive category is the category Ab itself. More precisely, Ab is a closed monoidal category. (Note tha...   » Read the article

Encyclopedia - Ideology: Encyclopedia Ii - Ideology - History Of The Concept Of Ideology
Perhaps the most accessible source for the original meaning of "ideology" is Hippolyte Taine's work on the Ancien Regime (first volume of...   » Read the article

Encyclopedia - Boolean Prime Ideal Theorem: Encyclopedia Ii - Boolean Prime Ideal Theorem - Prime Ideal Theorems
Before proceeding to actual prime ideal theorems, recall that an order theoretical ideal is just a (non-empty) directed lower set. If the...   » Read the article

Encyclopedia - Polynomial: Encyclopedia Ii - Polynomial - History
Determining the roots of polynomials, or "solving algebraic equations", is among the oldest problems in mathematics. Some polynomials, su...   » Read the article

Encyclopedia - Congruence Relation: Encyclopedia Ii - Congruence Relation - Universal Algebra
The idea is generalized in universal algebra: A congruence relation on an algebra A is a subset of the direct product A × A that is both...   » Read the article

Encyclopedia - Coprime: Encyclopedia Ii - Coprime - Generalizations
Two ideals A and B in the commutative ring R are called coprime if A + B = R. This generalizes Bézout's identity: with this definition, ...   » Read the article

Encyclopedia - Congruence Relation: Encyclopedia Ii - Congruence Relation - Linear Algebra
Two real matrices A and B are called congruent if there is an invertible real matrix P such that A symmetric matrix has real eigenvalu...   » Read the article

Encyclopedia - Congruence Relation: Encyclopedia Ii - Congruence Relation - Modular Arithmetic
The prototypical example is modular arithmetic: for n a positive natural number, two integers a and b are called congruent modulo n if a&...   » Read the article

Encyclopedia - Complete Lattice: Encyclopedia Ii - Complete Lattice - Representation
There are various other mathematical concepts that can be used to represent complete lattices. One means of doing so is the Dedekind-MacN...   » Read the article

Encyclopedia - Emanuel Lasker: Encyclopedia Ii - Emanuel Lasker - Mathematician
Lasker was also a distinguished mathematician. He performed his doctoral studies at Erlangen from 1900 to 1902 under David Hilbert. His d...   » Read the article

Encyclopedia - Emanuel Lasker: Encyclopedia Ii - Emanuel Lasker - Other Facets Of His Life
He was also a philosopher, and a good friend of Albert Einstein. Later in life he became an ardent humanitarian, and wrote passionately a...   » Read the article

Encyclopedia - Heian Period: Encyclopedia Ii - Heian Period - History
The Heian period is preceded by the Nara period and began in 794 after the movement of the capital of Japanese civilization to Heian-kyō...   » Read the article

Encyclopedia - Filter Mathematics: Encyclopedia Ii - Filter Mathematics - General Definition
A non-empty subset F of a partially ordered set (P,≤) is a filter, if the following conditions hold: For every x, y in F, there is som...   » Read the article

Encyclopedia - Field Mathematics: Encyclopedia Ii - Field Mathematics - Introduction
Fields are important objects of study in algebra since they provide a useful generalization of many number systems, such as the rational ...   » Read the article

Encyclopedia - Equivalence Of Categories: Encyclopedia Ii - Equivalence Of Categories - Definition
Formally, given two categories C and D, an equivalence of categories consists of a functor F : C -> D, a functor G : D ->...   » Read the article

Encyclopedia - Complete Lattice: Encyclopedia Ii - Complete Lattice - Morphisms Of Complete Lattices
The traditional morphisms between complete lattices are the complete homomorphisms (or complete lattice homomorphisms). These are charact...   » Read the article

Encyclopedia - Christian Left: Encyclopedia Ii - Christian Left - Notable Christian Leftists
Christian left - Australia. Frank Brennan Jesuit and advocate for Australia's Indigenous peoples Tim Costello Baptist minister Peter G...   » Read the article

Encyclopedia - Meaning Of Life: Encyclopedia Ii - Meaning Of Life - Spiritual Views
Mitch Albom wrote about his dying professor Morrie and their last lessons together in the bestseller Tuesdays with Morrie in which some i...   » Read the article

Encyclopedia - Meaning Of Life: Encyclopedia Ii - Meaning Of Life - Religious Beliefs
It is often suggested that religion is a response to humanity's need to simply stop being confused, or to the fear of death (and the conc...   » Read the article

Encyclopedia - Meaning Of Life: Encyclopedia Ii - Meaning Of Life - Scientific Approaches And Theories
Where scientists and philosophers converge on the quest for the meaning of life is an assumption that the mechanics of life (i.e., the un...   » Read the article

Encyclopedia - Meaning Of Life: Encyclopedia Ii - Meaning Of Life - Popular Beliefs
"What is the meaning of life?" is a question many people ask themselves at some point during their lifetime. Some people believe that the...   » Read the article

Encyclopedia - Meaning Of Life: Encyclopedia Ii - Meaning Of Life - Humorous Treatments
The very concept "the meaning of life" has become such a cliché that it has often been parodied, such as in the radio series The Hitchhi...   » Read the article

Encyclopedia - Body Image: Encyclopedia Ii - Body Image - Research: Measuring Body Image
Body image is often measured by asking the subject to rate his or her current and ideal body shape using a series of drawings of increasi...   » Read the article

Encyclopedia - Christian Left: Encyclopedia Ii - Christian Left - Christian Left In The United States
In the United States, members of the Christian Left come from a spectrum of denominations: Peace churches, elements of the Protestant mai...   » Read the article

Encyclopedia - Body Image: Encyclopedia Ii - Body Image - Attractiveness And Social Issues
Some researchers also found that men judge the female figure they found most attractive as heavier than women's ratings of the ideal body...   » Read the article

Encyclopedia - Body Image: Encyclopedia Ii - Body Image - Relationship To Psychological Disorders
Poor body images can often contribute to the onset of a variety of eating disorders, including anorexia nervosa, bulimia, and binge eatin...   » Read the article

Encyclopedia - Body Image: Encyclopedia Ii - Body Image - Causes And Influences
As clearly evidenced by literature around the world, poor body images have existed at least since the widespread availability of mirrors,...   » Read the article

Encyclopedia - Zorn's Lemma: Encyclopedia Ii - Zorn's Lemma - Sketch Of The Proof Of Zorn's Lemma
A sketch of the proof of Zorn's lemma follows. Suppose the lemma is false. Then there exists a partially ordered set, or poset, P such th...   » Read the article

Encyclopedia - Heian Period: Encyclopedia Ii - Heian Period - Developments In Buddhism
Buddhism began to spread throughout Japan during the Heian period, primarily through two major esoteric sects, Tendai (Heavenly Terrace) ...   » Read the article