In mathematics, ideal theory is the theory of ideals in commutative rings; and is the precursor name for the contemporary subject of commutative algebra. The name grew out of the central considerations, such as the Lasker–Noether theorem in algebraic geometry, and the ideal class group in algebraic number theory, of the commutative algebra of the first quarter of the twentieth century. It was used in the influential van der Waerden text on abstract algebra from around 1930.
The ideal theory in question had been based on elimination theory, but in line with David Hilbert's taste moved away from algorithmic methods. Gröbner basis theory has now reversed the trend, for computer algebra.
The importance of the ideal in general of a module, more general than an ideal, probably led to the perception that ideal theory was too narrow a description. Valuation theory, too, was an important technical extension, and was used by Helmut Hasse and Oscar Zariski. Bourbaki used commutative algebra; sometimes local algebra is applied to the theory of local rings. D. G. Northcott's 1953 Cambridge Tract Ideal Theory (reissued 2004 under the same title) was one of the final appearances of the name.
Read more about Ideal Theory: In Political Philosophy
Famous quotes containing the words ideal and/or theory:
“I’m no idealist to believe firmly in the integrity of our courts and in the jury system—that is no ideal to me, it is a living, working reality. Gentlemen, a court is no better than each man of you sitting before me on this jury. A court is only as sound as its jury, and a jury is only as sound as the men who make it up.”
—Harper Lee (b. 1926)
“[Anarchism] is the philosophy of the sovereignty of the individual. It is the theory of social harmony. It is the great, surging, living truth that is reconstructing the world, and that will usher in the Dawn.”
—Emma Goldman (1869–1940)