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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, specifically
commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...
, a proper
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considered ...
''Q'' of a
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
''A'' is said to be primary if whenever ''xy'' is an element of ''Q'' then ''x'' or ''y''''n'' is also an element of ''Q'', for some ''n'' > 0. For example, in the
ring of integers In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often deno ...
Z, (''p''''n'') is a primary ideal if ''p'' is a prime number. The notion of primary ideals is important in commutative ring theory because every ideal of a
Noetherian ring In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noethe ...
has a
primary decomposition In mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be decomposed as an intersection, called primary decomposition, of finitely many '' primary ideals'' (which are relate ...
, that is, can be written as an intersection of finitely many primary ideals. This result is known as the Lasker–Noether theorem. Consequently, an irreducible ideal of a Noetherian ring is primary. Various methods of generalizing primary ideals to noncommutative rings exist, but the topic is most often studied for commutative rings. Therefore, the rings in this article are assumed to be commutative rings with identity.


Examples and properties

* The definition can be rephrased in a more symmetric manner: an ideal \mathfrak is primary if, whenever x y \in \mathfrak, we have x \in \mathfrak or y \in \mathfrak or x, y \in \sqrt. (Here \sqrt denotes the
radical Radical may refer to: Politics and ideology Politics * Radical politics, the political intent of fundamental societal change *Radicalism (historical), the Radical Movement that began in late 18th century Britain and spread to continental Europe an ...
of \mathfrak.) * An ideal ''Q'' of ''R'' is primary if and only if every
zero divisor In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right ze ...
in ''R''/''Q'' is nilpotent. (Compare this to the case of prime ideals, where ''P'' is prime if and only if every zero divisor in ''R''/''P'' is actually zero.) * Any
prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together wi ...
is primary, and moreover an ideal is prime if and only if it is primary and
semiprime In mathematics, a semiprime is a natural number that is the product of exactly two prime numbers. The two primes in the product may equal each other, so the semiprimes include the squares of prime numbers. Because there are infinitely many prime ...
(also called
radical ideal In ring theory, a branch of mathematics, the radical of an ideal I of a commutative ring is another ideal defined by the property that an element x is in the radical if and only if some power of x is in I. Taking the radical of an ideal is called ' ...
in the commutative case). * Every primary ideal is primal. * If ''Q'' is a primary ideal, then the
radical Radical may refer to: Politics and ideology Politics * Radical politics, the political intent of fundamental societal change *Radicalism (historical), the Radical Movement that began in late 18th century Britain and spread to continental Europe an ...
of ''Q'' is necessarily a prime ideal ''P'', and this ideal is called the
associated prime ideal In abstract algebra, an associated prime of a module ''M'' over a ring ''R'' is a type of prime ideal of ''R'' that arises as an annihilator of a (prime) submodule of ''M''. The set of associated primes is usually denoted by \operatorname_R(M), ...
of ''Q''. In this situation, ''Q'' is said to be ''P''-primary. ** On the other hand, an ideal whose radical is prime is not necessarily primary: for example, if R = k ,y,z(x y - z^2), \mathfrak = (\overline, \overline), and \mathfrak = \mathfrak^2, then \mathfrak is prime and \sqrt = \mathfrak, but we have \overline \overline = ^2 \in \mathfrak^2 = \mathfrak, \overline \not \in \mathfrak, and ^n \not \in \mathfrak for all n > 0, so \mathfrak is not primary. The primary decomposition of \mathfrak is (\overline) \cap (^2, \overline \overline, \overline); here (\overline) is \mathfrak-primary and (^2, \overline \overline, \overline) is (\overline, \overline, \overline)-primary. *** An ideal whose radical is ''maximal'', however, is primary. *** Every ideal with radical ''is'' contained in a smallest -primary ideal: all elements such that for some . The smallest -primary ideal containing is called the th
symbolic power The concept of symbolic power, also known as symbolic domination (''domination symbolique'' in French language) or symbolic violence, was first introduced by French sociologist Pierre Bourdieu to account for the tacit, almost unconscious modes of ...
of . * If ''P'' is a maximal prime ideal, then any ideal containing a power of ''P'' is ''P''-primary. Not all ''P''-primary ideals need be powers of ''P'', but at least they contain a power of P; for example the ideal (''x'', ''y''2) is ''P''-primary for the ideal ''P'' = (''x'', ''y'') in the ring ''k'' 'x'', ''y'' but is not a power of ''P'', however it contains P². * If ''A'' is a
Noetherian ring In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noethe ...
and ''P'' a prime ideal, then the kernel of A \to A_P, the map from ''A'' to the localization of ''A'' at ''P'', is the intersection of all ''P''-primary ideals. * A finite nonempty product of \mathfrak-primary ideals is \mathfrak-primary but an infinite product of \mathfrak-primary ideals may not be \mathfrak p-primary; since for example, in a Noetherian local ring with maximal ideal \mathfrak m, \cap_ \mathfrak^n = 0 ( Krull intersection theorem) where each \mathfrak^n is \mathfrak-primary, for example the infinite product of the maximal (and hence prime and hence primary) ideal m=\langle x,y \rangle of the local ring K ,y\langle x^2, xy\rangle yields the zero ideal, which in this case is not primary (because the zero divisor y is not nilpotent). In fact, in a Noetherian ring, a nonempty product of \mathfrak-primary ideals Q_i is \mathfrak-primary if and only if there exists some integer n > 0 such that \mathfrak^n \subset \cap_i Q_i.


Footnotes


References

* *Bourbaki, ''Algèbre commutative''. * * *
On primal ideals
Ladislas Fuchs *{{citation , author1=Lesieur, L. , author2=Croisot, R. , title=Algèbre noethérienne non commutative , language=French , publisher=Mémor. Sci. Math., Fasc. CLIV. Gauthier-Villars & Cie, Editeur -Imprimeur-Libraire, Paris , year=1963 , pages=119 , mr=0155861


External links


''Primary ideal'' at Encyclopaedia of Mathematics
Commutative algebra Ideals (ring theory)