Additive theory of ideals

A branch of modern algebra. Its principal task is to represent any ideal of a ring (or of another algebraic system) as the intersection of a finite number of ideals of special type (primary, tertiary, primal, uniserial, etc.). The type of the representation is so chosen that: 1) for any ideal there exists a representation, in other words, some "existence" theorem holds; 2) the representations chosen must be unique apart from certain limitations or, in other words, some "uniqueness" theorem must hold. The fundamental principles of the additive theory of ideals were introduced in the 1920s and the 1930s by E. Noether [1] and W. Krull [2].

All special features of the additive theory of ideals are clearly manifested in the case of rings. Let $ R $ be a Noetherian ring, i.e. an associative ring with the maximum condition for ideals. If $ A $ is an ideal of $ R $, then there exists a largest ideal $ N $ of $ R $ for which $ N ^ {k} \subseteq A $ for some integer $ k \geq 1 $. This ideal $ N $ is known as the primary radical of $ A $( in $ R $) and is denoted by $ \mathop{\rm pr} (A) $. An ideal $ Q $ of $ R $ is said to be primary if for any two ideals $ A $ and $ B $ of $ R $, the condition

$$ AB \subseteq Q , A \nsubseteq Q \Rightarrow B \subseteq \mathop{\rm pr} ( Q ) $$

is satisfied. The intersection theorem is valid for primary ideals: The intersection of two primary ideals having the same primary radical $ P $ is itself a primary ideal with radical $ P $. This theorem is used to prove an existence theorem: If the ring $ R $ is commutative, then for any ideal $ A \neq R $ there exists a representation of $ A $ as the intersection of a finite number of primary ideals $ A _ {i} $:

$$ \tag{1 } A = A _ {1} \cap \dots \cap A _ {n} , $$

such that none of the ideals $ A _ {i} $ contains the intersection of the other ones, and such that the primary radicals $ \mathop{\rm pr} ( A _ {i} ) $ are pairwise different. Such representations are known as non-contractible or primarily reduced [1], [4]. The uniqueness theorem holds for such representations: If (1) holds and

$$ \tag{2 } A = B _ {1} \cap \dots \cap B _ {m} $$

is a second primarily-reduced representation of the ideal $ A $ of the ring $ R $, then $ m = n $ and $ \mathop{\rm pr} ( A _ {i} ) = \mathop{\rm pr} ( B _ {i} ) $ for $ 1 \leq i \leq n $, provided the ideals $ B _ {i} $ are suitably renumbered.

The additive theory of ideals of Noetherian commutative rings (the classical additive theory of ideals) has found numerous applications in various branches of mathematics.

If the ring $ R $ is non-commutative, the above-mentioned existence theorem is no longer valid, but the uniqueness and intersection theorems still hold. This is why, ever since the 1930s, repeated attempts have been made to find a generalization of classical primarity to the non-commutative case such that the existence theorem, too, remains valid. Such a generalization has in fact been found [4], and is known as tertiarity (cf. Tertiary ideal). It was subsequently shown that, within certain natural limitations, tertiarity is the only "good" generalization of the concept of primarity [6], [7], [8].

During the 1960s the additive theory of ideals developed further within the framework of lattice theory, of systems with fractions and of multiplicative systems [4], [5], [6]; this stimulated the development of the additive theory of ideals for non-associative rings, normal divisors of a group and submodules of a module.

References

[1]	E. Noether, "Idealtheorie in Ringbereichen" Math. Ann. , 83 (1921) pp. 24–66
[2]	W. Krull, "Idealtheorie in Ringen ohne Endlichkeitsbedingung" Math. Ann. , 101 (1929) pp. 729–744
[3]	O. Zariski, P. Samuel, "Commutative algebra" , 1 , Springer (1975)
[4]	L. Lesieur, R. Croisot, "Algèbre noethérienne noncommutative" , Gauthier-Villars (1963)
[5]	K. Murata, "Additive ideal theory in multiplicative systems" J. Inst. Polytechn. Osaka City Univ. (A) , 10 : 2 (1959) pp. 91–115
[6]	V.A. Andrunakievich, Yu.M. Ryabukhin, "The additive theory of ideals in systems with residuals" Math. USSR-Izv. , 1 : 5 (1967) pp. 1011–1040 Izv. Akad. Nauk SSSR Ser. Mat. , 31 (1967) pp. 1057–1090
[7]	J.A. Riley, "Axiomatic primary and tertiary decomposition theory" Trans. Amer. Math. Soc. , 105 (1962) pp. 177–201
[8]	I.M. Goyan, Yu.M. Ryabukhin, "On the axiomatic additive theory of Riley ideals" Mat. Issl. , 2 : 1 (1967) pp. 14–25 (In Russian)
[9]	L. Fuchs, "On primal ideals" Proc. Amer. Math. Soc. , 1 (1950) pp. 1–6
[10]	Itogi Nauk. Algebra Topol. Geom. 1965 (1967) pp. 133–180

Comments

A primary representation is also called a primary decomposition. More generally one has primary decompositions for submodules $ N $ of modules $ M $ over a Noetherian ring $ R $. This is a representation

$$ N = \cap _ { i } Q _ {i} $$

where each $ \mathop{\rm Ass} ( M / Q _ {i} ) $ consists of a single prime ideal $ \mathfrak p _ {i} $. (By definition $ \mathop{\rm Ass} ( M ) $ for a module $ M $, the set of prime ideals associated to $ M $, is the collection of all prime ideals $ \mathfrak p $ for which there exists an $ x \in M $ such that $ \mathfrak p = \{ {r \in R } : {r x = 0 } \} $). There is also a corresponding uniqueness theorem, stating that there is a reduced decomposition, which of course means that $ \cap _ {j \neq i } Q _ {j} \subset Q _ {i} $ holds for no $ i \in I $ and that the $ \mathop{\rm Ass} ( M / Q _ {i} ) = \mathfrak p _ {i} $ are all different.

References

[a1]	N. Bourbaki, "Algèbre commutative" , Hermann (1961) pp. Chapt. 3; 4