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In algebra and algebraic geometry, given a commutative Noetherian ring and an ideal in it, the n-th symbolic power of is the ideal = (/) ()where is the localization of at , we set : is the canonical map from a ring to its localization, and the intersection runs through all of the associated primes of /.
A ring is a prime ring if and only if the zero ideal is a prime ideal, and moreover a ring is a domain if and only if the zero ideal is a completely prime ideal. Another fact from commutative theory echoed in noncommutative theory is that if A is a nonzero R - module , and P is a maximal element in the poset of annihilator ideals of submodules ...
An ideal whose radical is maximal, however, is primary. Every ideal Q with radical P is contained in a smallest P-primary ideal: all elements a such that ax ∈ Q for some x ∉ P. The smallest P-primary ideal containing P n is called the n th symbolic power of P. If P is a maximal prime ideal, then any ideal containing a power of P is P-primary.
Let be a Noetherian ring, x an element of it and a minimal prime over x.Replacing A by the localization, we can assume is local with the maximal ideal .Let be a strictly smaller prime ideal and let () =, which is a -primary ideal called the n-th symbolic power of .
Primary ideal: An ideal I is called a primary ideal if for all a and b in R, if ab is in I, then at least one of a and b n is in I for some natural number n. Every prime ideal is primary, but not conversely. A semiprime primary ideal is prime. Principal ideal: An ideal generated by one element. [15]
Thus, in the setting of preceding theorem, the primary ideal Q corresponding to a minimal prime P is also the smallest P-primary ideal containing I and is called the P-primary component of I. For example, if the power P n of a prime P has a primary decomposition, then its P-primary component is the n-th symbolic power of P.
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A radical ideal (or semiprime ideal) is an ideal that is equal to its radical. The radical of a primary ideal is a prime ideal . This concept is generalized to non-commutative rings in the semiprime ring article.