Ideal theory

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. 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. Let R be a ring and M an R-module. Then each ideal a {displaystyle {mathfrak {a}}} of R determines a topology on M called the a {displaystyle {mathfrak {a}}} -adic topology such that a subset U of M is open if and only if for each x in U there exists a positive integer n such that With respect to this a {displaystyle {mathfrak {a}}} -adic topology, the module operations are continuous; in particular, M {displaystyle M} is a possibly non-Hausdorff topological group. Also, M is a Hausdorff topological space if and only if ⋂ n > 0 a n M = 0. { extstyle igcap _{n>0}{mathfrak {a}}^{n}M=0.} Moreover, when M {displaystyle M} is Hausdorff, the topology is the same as the metric space topology given by defining the distance function: d ( x , y ) = 2 − n {displaystyle d(x,y)=2^{-n}} for x ≠ y {displaystyle x eq y} , where n {displaystyle n} is an integer such that x − y ∈ a n M − a n + 1 M {displaystyle x-yin {mathfrak {a}}^{n}M-{mathfrak {a}}^{n+1}M} . Given a submodule N of M, the a {displaystyle {mathfrak {a}}} -closure of N in M is equal to ⋂ n > 0 ( N + a n M ) { extstyle igcap _{n>0}(N+{mathfrak {a}}^{n}M)} , as shown easily. Now, a priori, on a submodule N of M, there are two natural a {displaystyle {mathfrak {a}}} -topologies: the subspace topology induced by the a {displaystyle {mathfrak {a}}} -adic topology on M and the a {displaystyle {mathfrak {a}}} -adic topology on N. However, when R {displaystyle R} is Noetherian and M {displaystyle M} is finite over it, those two topologies coincide as a consequence of the Artin–Rees lemma. When M {displaystyle M} is Hausdorff, M {displaystyle M} can be completed as a metric space; the resulting space is denoted by M ^ {displaystyle {widehat {M}}} and has the module structure obtained by extending the module operations by continuity. It is also the same as (or canonically isomorphic to): where the right-hand side is the completion of the module M {displaystyle M} with respect to a {displaystyle {mathfrak {a}}} .