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On the embedded primary components of ideals. IV

1995; American Mathematical Society; Volume: 347; Issue: 2 Linguagem: Inglês

10.1090/s0002-9947-1995-1249882-7

1088-6850

William Heinzer, Louis J. Ratliff, Kishor Shah,

Rings, Modules, and Algebras

The results in this paper expand the fundamental decomposition theory of ideals pioneered by Emmy Noether. Specifically, let I I be an ideal in a local ring ( R , M ) (R,M) that has M M as an embedded prime divisor, and for a prime divisor P P of I I let I C P ( I ) I{C_P}(I) be the set of irreducible components q q of I I that are P P -primary (so there exists a decomposition of I I as an irredundant finite intersection of irreducible ideals that has q q as a factor). Then the main results show: (a) I C M ( I ) = ∪ { I C M ( Q ) ; Q is a MEC ⁡ of I } I{C_M}(I) = \cup \{ I{C_M}(Q);Q\;{\text {is a }}\operatorname {MEC} {\text { of }}I\} ( Q Q is a MEC of I I in case Q Q is maximal in the set of M M -primary components of I I ); (b) if I = ∩ { q i ; i = 1 , … , n } I = \cap \{ {q_i};i = 1, \ldots ,n\} is an irredundant irreducible decomposition of I I such that q i {q_i} is M M -primary if and only if i = 1 , … , k > n i = 1, \ldots ,k > n , then ∩ { q i ; i = 1 , … , k } \cap \{ {q_i};i = 1, \ldots ,k\} is an irredundant irreducible decomposition of a MEC of I I , and, conversely, if Q Q is a MEC of I I and if ∩ { Q j ; j = 1 , … , m } \cap \{ {Q_j};j = 1, \ldots ,m\} (resp., ∩ { q i ; i = 1 , … , n } \cap \{ {q_i};i = 1, \ldots ,n\} ) is an irredundant irreducible decomposition of Q Q (resp., I I ) such that q 1 , … , q k {q_1}, \ldots ,{q_k} are the M M -primary ideals in { q 1 , … , q n } \{ {q_1}, \ldots ,{q_n}\} , then m = k m = k and ( ∩ { q i ; i = k + 1 , … , n } ) ∩ ( ∩ { Q j ; j = 1 , … , m } ) ( \cap \{ {q_i};i = k + 1, \ldots ,n\} ) \cap ( \cap \{ {Q_j};j = 1, \ldots ,m\} ) is an irredundant irreducible decomposition of I I ; (c) I C M ( I ) = { q , q is maximal in the set of ideals that contain I and do not contain I : M } I{C_M}(I) = \{ q,q\;{\text {is maximal in the set of ideals that contain }}I\;{\text {and do not contain }}I:M\} ; (d) if Q Q is a MEC of I I , then I C M ( Q ) = { q ; Q ⊆ q ∈ I C M ( I ) } I{C_M}(Q) = \{ q;Q \subseteq q \in I{C_M}(I)\} ; (e) if J J is an ideal that lies between I I and an ideal Q ∈ I C M ( I ) Q \in I{C_M}(I) , then J = ∩ { q ; J ⊆ q ∈ I C M ( I ) } J = \cap \{ q;J \subseteq q \in I{C_M}(I)\} ; and, (f) there are no containment relations among the ideals in ∪ { I C P ( I ) \cup \{ I{C_P}(I) ; P P is a prime divisor of I I }.