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Concept
Ideal theory
Summary
In mathematics, ideal theory is the theory of ideals in commutative rings. While the notion of an ideal exists also for non-commutative rings, a much more substantial theory exists only for commutative rings (and this article therefore only considers ideals in commutative rings.)
Throughout the articles, rings refer to commutative rings. See also the article ideal (ring theory) for basic operations such as sum or products of ideals.
finitely generated algebra
Ideals in a finitely generated algebra over a field (that is, a quotient of a polynomial ring over a field) behave somehow nicer than those in a general commutative ring. First, in contrast to the general case, if is a finitely generated algebra over a field, then the radical of an ideal in is the intersection of all maximal ideals containing the ideal (because is a Jacobson ring). This may be thought of as an extension of Hilbert's Nullstellensatz, which concerns the case when is a polynomial ring.
I-adic topology
If I is an ideal in a ring A, then it determines the topology on A where a subset U of A is open if, for each x in U,
for some integer . This topology is called the I-adic topology. It is also called an a-adic topology if is generated by an element .
For example, take , the ring of integers and an ideal generated by a prime number p. For each integer , define when , prime to . Then, clearly,
where denotes an open ball of radius with center . Hence, the -adic topology on is the same as the metric space topology given by . As a metric space, can be completed. The resulting complete metric space has a structure of a ring that extended the ring structure of ; this ring is denoted as and is called the ring of p-adic integers.
In a Dedekind domain A (e.g., a ring of integers in a number field or the coordinate ring of a smooth affine curve) with the field of fractions , an ideal is invertible in the sense: there exists a fractional ideal (that is, an A-submodule of ) such that , where the product on the left is a product of submodules of K.
Official source
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