The difference between Ideal and Principal ideal
When used as nouns, ideal means a perfect standard of beauty, intellect etc., or a standard of excellence to aim at, whereas principal ideal means an ideal i in an algebraic object r (which could be a ring, algebra, semigroup or lattice) that is generated by a given single element a ∈ r.
Ideal is also adjective with the meaning: optimal.
check bellow for the other definitions of Ideal and Principal ideal
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Ideal as an adjective:
Optimal; being the best possibility.
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Ideal as an adjective:
Perfect, flawless, having no defects.
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Ideal as an adjective:
Pertaining to ideas, or to a given idea.
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Ideal as an adjective:
Existing only in the mind; conceptual, imaginary.
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Ideal as an adjective:
Teaching or relating to the doctrine of idealism.
Examples:
"the ideal theory or philosophy"
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Ideal as an adjective (mathematics):
Not actually present, but considered as present when limits at infinity are included.
Examples:
"'ideal point"
"An ideal triangle in the hyperbolic disk is one bounded by three geodesics that meet precisely on the circle."
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Ideal as a noun:
A perfect standard of beauty, intellect etc., or a standard of excellence to aim at.
Examples:
"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'' - [[w:Carl Schurz Carl Schurz]]"
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Ideal as a noun (algebra, ring theory):
A subring closed under multiplication by its containing ring.
Examples:
"Let <math>\mathbb{Z}</math> be the ring of integers and let <math>2\mathbb{Z}</math> be its ideal of even integers. Then the quotient ring <math>\mathbb{Z} / 2\mathbb{Z}</math> is a Boolean ring."
"The product of two ideals <math>\mathfrak{a}</math> and <math>\mathfrak{b}</math> is an ideal <math>\mathfrak{a b}</math> which is a subset of the intersection of <math>\mathfrak{a}</math> and <math>\mathfrak{b}</math>. This should help to understand why maximal ideals are prime ideals. Likewise, the union of <math>\mathfrak{a}</math> and <math>\mathfrak{b}</math> is a subset of <math>\mathfrak{a + b}</math>."
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Ideal as a noun (algebra, order theory, lattice theory):
A non-empty lower set (of a partially ordered set) which is closed under binary suprema (a.k.a. joins).
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Ideal as a noun (set theory):
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:
"Formally, an ideal <math>I</math> of a given set <math>X</math> is a nonempty subset of the [[powerset]] <math>\mathcal{P}(X)</math> such that: <math>(1)\ \emptyset \in I</math>, <math>(2)\ A \in I \and B \subseteq A\implies B\in I</math> and <math>(3)\ A,B \in I\implies A\cup B \in I</math>."
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Ideal as a noun (algebra, Lie theory):
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.
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Principal ideal as a noun (algebra):
An ideal I in an algebraic object R (which could be a ring, algebra, semigroup or lattice) that is generated by a given single element a ∈ R; the smallest ideal that contains a.