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
Other articles related to "ideal theory, ideal, theory":
... In political philosophy, ideal theory refers to argument concerning political or social arrangements under favorable assumptions ...
... First, it is a set of claims about the features of an ideal scientific theory an ideal theory is the sort of theory science aims to produce ... it is the commitment that science will eventually produce theories very much like an ideal theory and that science has done pretty well thus far in some domains ... According to scientific realism, an ideal scientific theory has the following features The claims the theory makes are either true or false, depending on whether the entities talked about by the theory exist and are ...
... Fractional ideal Ideal class group Radical of an ideal Hilbert's Nullstellensatz. ...
... The content-giving part of Rawls' thesis belongs to Ideal Theory, it is an attempt to define how different peoples who are just, or at least decent, should behave with respect to one another ... Rawls refers to this ideal conception as a "realistic utopia" realistic because it could and may exist utopian because it "joins reasonableness and justice with conditions ... It is typical of Rawls's approach that he focuses on ideal theory and does not discuss to any great extent non-ideal theory, which involves considering the proper response to injustice ...
Famous quotes containing the words theory and/or ideal:
“every subjective phenomenon is essentially connected with a single point of view, and it seems inevitable that an objective, physical theory will abandon that point of view.”
—Thomas Nagel (b. 1938)
“The unique eludes us; yet we remain faithful to the ideal of it; and in spite of sense and of our merely abstract thinking, it becomes for us the most real thing in the actual world, although for us it is the elusive goal of an infinite quest.”
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