Primary ideal
of a commutative ring
An ideal such that if
and
, then either
or
for some natural number
. In the ring
of integers a primary ideal is an ideal of the form
, where
is a prime number and
is a natural number. In commutative algebra an important role is played by the representation of an arbitrary ideal of a commutative Noetherian ring as an intersection of a finite number of primary ideals — a primary decomposition. More generally, let
denote the set of prime ideals of a ring
that are annihilators of non-zero submodules of a module
. A submodule
of a module
over a Noetherian ring
is called primary if
is a one-element set. If
is commutative, then every proper submodule of a Noetherian
-module that cannot be represented as an intersection of submodules strictly containing it is primary. In the non-commutative case this is not true and therefore attempts have been undertaken to construct various non-commutative generalizations of the notion of primarity. E.g., a proper submodule
of a module
is called primary if for every non-zero injective submodule
of the injective hull
of the module
(cf. Injective module) the intersection of the kernels of the homomorphisms from
into
is trivial. Another successful generalization is the notion of a tertiary ideal [4]: A left ideal
of a left Noetherian ring
is called tertiary if, for any
,
, it follows from
that, for any
, there is an element
such that
. Both these generalizations lead to a non-commutative analogue of primary decomposition. Every tertiary ideal of a Noetherian ring
is primary if and only if
satisfies the Artin–Rees condition: For arbitrary left ideals
of
there is a natural number
such that
(cf. [3]).
References
[1] | N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French) |
[2] | O. Zariski, P. Samuel, "Commutative algebra" , 1 , Springer (1975) |
[3] | O. Goldman, "Rings and modules of quotients" J. of Algebra , 13 : 1 (1969) pp. 10–47 |
[4] | L. Lesieur, R. Croisot, "Algèbre noethérienne noncommutative" , Gauthier-Villars (1963) |
Primary ideal. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Primary_ideal&oldid=42013