ideal
Language: English
Part of speech: noun
Pronunciation: ipa: /aɪˈdɪəl/
ipa: /aɪˈdi(ə)l/
note: Indic
ipa: /ˈajɖɪjəl/
rhymes: -iːəl
Forms: form: ideals
Etymology: From French idéal, from Late Latin ideālis (“existing in idea”), by surface analysis, idea + -al, from Latin idea (“idea”); see idea.
In mathematics, the noun ring theory sense was first introduced by German mathematician Richard Dedekind in his 1871 edition of a text on number theory. The concept was quickly expanded to ring theory and later generalised to order theory. The set theory and Lie theory senses can be regarded as applications of the order theory sense.
Senses: examples: text: Ideals are like stars; you will not succeed in touching them with your hands. But like the seafaring man on the desert of waters, you choose them as your guides, and following them you will reach your destiny - Carl Schurz
text: With great humility, I call upon all Americans to help me keep our nation united in defense of those ideals which have been so eloquently proclaimed by Franklin Roosevelt. I want in turn to assure my fellow Americans and all of those who love peace and liberty throughout the world that I will support and defend those ideals with all my strength and all my heart.
ref: 1945 April 16, Harry S. Truman, 9:21 from the start, in MP72-20 President Roosevelt’s Funeral and Procession; Truman – New President of U.S., Harry S. Truman Presidential Library and Museum, National Archives Identifier: 595162:
glosses: A perfect standard of beauty, intellect etc., or a standard of excellence to aim at.
examples: text: Let #92;mathbb#123;Z#125; be the ring of integers and let 2#92;mathbb#123;Z#125; be its ideal of even integers. Then the quotient ring #92;mathbb#123;Z#125;#47;2#92;mathbb#123;Z#125; is a Boolean ring.
text: The product of two ideals #92;mathfrak#123;a#125; and #92;mathfrak#123;b#125; is an ideal #92;mathfrak#123;ab#125; which is a subset of the intersection of #92;mathfrak#123;a#125; and #92;mathfrak#123;b#125;. This should help to understand why maximal ideals are prime ideals. Likewise, the union of #92;mathfrak#123;a#125; and #92;mathfrak#123;b#125; is a subset of #92;mathfrak#123;a#43;b#125;.
text: In trying to understand the ideal theory of a commutative ring, one quickly sees that it is important to first understand the prime ideals.
ref: 2004, K. R. Goodearl; R. B. Warfield, Jr., An Introduction to Noncommutative Noetherian Rings, 2nd edition, Cambridge University Press, page 47:
text: If an ideal I of a ring contains the multiplicative identity 1, then we have seen that I must be the entire ring.
ref: 2009, John J. Watkins, Topics in Commutative Ring Theory, Princeton University Press, page 45:
text: However, every R has a minimal prime ideal consisting of left zero-divisors and one of right zero-divisors.
ref: 2010, W. D. Burgess, A. Lashgari, A. Mojiri, “Elements of Minimal Prime Ideals in General Rings”, in Sergio R. López-Permouth, Dinh Van Huynh, editors, Advances in Ring Theory, Springer (Birkhäuser), page 69:
topics: algebra
mathematics
sciences
qualifier: ring theory
glosses: A two-sided ideal; a subset of a ring which is closed under both left and right multiplication by elements of the ring.
examples: text: 1992, Unnamed translator, T. S. Fofanova, General Theory of Lattices, in Ordered Sets and Lattices II, American Mathematical Society, page 119,
An ideal A of L is called complete if it contains all least upper bounds of its subsets that exist in L. Bishop and Schreiner [80] studied conditions under which joins of ideals in the lattices of all ideals and of all complete ideals coincide.
text: 1.35 Find a distributive lattice L with no minimal and no maximal prime ideals.
ref: 2011, George Grätzer, Lattice Theory: Foundation, Springer (Birkhäuser), page 125:
text: Definition 15.11 (Width Ideal) An ideal Q of a poset P = (X,≤) is a width ideal if maximal(Q) is a width antichain.
ref: 2015, Vijay K. Garg, Introduction to Lattice Theory with Computer Science Applications, Wiley, page 186:
topics: algebra
mathematics
order-theory
sciences
qualifier: lattice theory
glosses: A non-empty lower set (of a partially ordered set) which is closed under binary suprema (a.k.a. joins).
examples: text: Formally, an ideal I of a given set X is a nonempty subset of the powerset #92;mathcal#123;P#125;(X) such that: (1)#92;#92;emptyset#92;inI, (2)#92;A#92;inI#92;andB#92;subseteqA#92;impliesB#92;inI and (3)#92;A,B#92;inI#92;impliesA#92;cupB#92;inI.
topics: mathematics
sciences
set-theory
glosses: A collection of sets, considered small or negligible, such that every subset of each member and the union of any two members are also members of the collection.
examples: text: If 𝖌 is a Lie algebra, 𝖍 is an ideal and the Lie algebras 𝖍 and 𝖌/𝖍 are solvable, then 𝖌 is solvable.
ref: 1975, Zhe-Xian Wan, translated by Che-Young Lee, Lie Algebras, Pergamon Press, page 13:
text: What really put primitive ideals in enveloping algebras of semisimple Lie algebras on the map was Duflo's fundamental theorem that any such ideal is the annihilator of a very special kind of simple module, namely a highest weight module.
ref: 2006, W. McGovern, “The work of Anthony Joseph in classical representation theory”, in Anthony Joseph, Joseph Bernstein, Vladimir Hinich, Anna Melnikov, editors, Studies in Lie Theory: Dedicated to A. Joseph on His Sixtieth Birthday, Springer (Birkhäuser), page 3:
text: Next let L be an arbitrary semisimple Lie algebra. Then L can be written uniquely as a direct sum L#95;1#92;oplus#92;dots#92;oplusL#95;t of simple ideals (Theorem 5.2).
ref: 2013, J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer, page 73:
topics: algebra
mathematics
sciences
qualifier: Lie theory
glosses: A Lie subalgebra (subspace that is closed under the Lie bracket) 𝖍 of a given Lie algebra 𝖌 such that the Lie bracket [𝖌,𝖍] is a subset of 𝖍.
examples: text: The set of natural numbers with multiplication as the monoid operation (instead of addition) has multiplicative ideals, such as, for example, the set {1, 3, 9, 27, 81, ...}. If any member of it is multiplied by a number which is not a power of 3 then the result will not be a power of three.
topics: algebra
mathematics
sciences
glosses: A subsemigroup with the property that if any semigroup element outside of it is added to any one of its members, the result must lie outside of it.
Synonyms: topics: order-theory
mathematics
sciences
word: order ideal
sense: type of Lie subalgebra
word: Lie ideal
Hyponyms: topics: mathematics
sciences
word: maximal ideal
topics: mathematics
sciences
word: principal ideal
Derived terms: factor ideal
fractional ideal
Frink ideal
ideal lattice
lattice ideal
left ideal
minimal ideal
Platonic ideal
primary ideal
prime ideal
principal ideal
principal ideal domain
principal ideal ring
pseudoideal
radical ideal
right ideal
subideal
two-sided ideal
vanishing ideal
zero ideal