Definition of "ideal"

There will always be a wide interval between practical and ideal excellence; […] .

1751 April 13, Samuel Johnson, The Rambler, Number 112, reprinted in 1825, The Works of Samuel Johnson, LL. D., Volume 1, Jones & Company, page 194

The idea of ghosts is ridiculous in the extreme; and if you continue to be swayed by ideal terrors —

1796, Matthew Lewis, The Monk, Folio Society, published 1985, page 256

Life and death appeared to me ideal bounds, which I should first break through, and pour a torrent of light into our dark world.

1818, [Mary Shelley], “Chapter 4”, in Frankenstein; or, The Modern Prometheus. […], volumes (please specify |volume=I to III), London: […] [Macdonald and Son] for Lackington, Hughes, Harding, Mavor, & Jones

At first, he began to doubt the reality of his adventures, but the acute pain in his shoulders when he attempted to rise, assured him that the kicking of the goblins was certainly not ideal.

1836 March – 1837 October, Charles Dickens, “(please specify the chapter name)”, in The Posthumous Papers of the Pickwick Club, London: Chapman and Hall, […], published 1837

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.

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

In trying to understand the ideal theory of a commutative ring, one quickly sees that it is important to first understand the prime ideals.

2004, K. R. Goodearl, R. B. Warfield, Jr., An Introduction to Noncommutative Noetherian Rings, 2nd edition, Cambridge University Press, page 47

If an ideal I of a ring contains the multiplicative identity 1, then we have seen that I must be the entire ring.

2009, John J. Watkins, Topics in Commutative Ring Theory, Princeton University Press, page 45

However, every R has a minimal prime ideal consisting of left zero-divisors and one of right zero-divisors.

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

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 studied conditions under which joins of ideals in the lattices of all ideals and of all complete ideals coincide.

1992, Unnamed translator, T. S. Fofanova, General Theory of Lattices, in Ordered Sets and Lattices II, American Mathematical Society, page 119

1.35 Find a distributive lattice L with no minimal and no maximal prime ideals.

2011, George Grätzer, Lattice Theory: Foundation‎, Springer (Birkhäuser), page 125

Definition 15.11 (Width Ideal) An ideal Q of a poset P = (X,≤) is a width ideal if maximal(Q) is a width antichain.

2015, Vijay K. Garg, Introduction to Lattice Theory with Computer Science Applications, Wiley, page 186

If 𝖌 is a Lie algebra, 𝖍 is an ideal and the Lie algebras 𝖍 and 𝖌/𝖍 are solvable, then 𝖌 is solvable.

1975, Zhe-Xian Wan, translated by Che-Young Lee, Lie Algebras, Pergamon Press, page 13

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.

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

Next let L {\displaystyle L} be an arbitrary semisimple Lie algebra. Then L {\displaystyle L} can be written uniquely as a direct sum L 1 ⊕ ⋯ ⊕ L t {\displaystyle L_{1}\oplus \dots \oplus L_{t}} of simple ideals (Theorem 5.2).

2013, J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer, page 73