By Max D. Larsen, Paul J. McCarthy
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Example text
6, we obtain the following result. 7. Theorem. If the R-module M satisjies (ACC), then every sub- module of M can be written as an intersection of a finite number of primary submodules. An ideal Q of a ring R is called primary if it is a primary submodule of R when considered as an R-module. Since R has a unity, Q is a primary ideal if and only if for all a, b E R, ub E Q and b # Q imply that an E Q for some positive integer n. Since R is commutative we shall call it simply Noetherian if it is left Noetherian.
X , ] . 19. Theorem (Artin-Rees Lemma). Let R be a Noetherian ring and let A, B, and C be ideals of R. Then there is a positive integer r such that AnB n C = An-'(ArB n C ) for all n > r. Proof. Let A = (al, .. Let X , , . , X, be indeterminates and let S , be the set of all homogeneous polynomials f E R IX l , . . , xk] of degree n such that f ( a l , . . , ak) E A " B n C. ) Let = uFz0S, and let A' be the ideal of R[X,, . , xk] generated by S. 17, A' is finitely generated, say A' = (fl, .
Therefore Q = (Q a"M) n (Q Rx). Thus anM c Q. We conclude that Q is primary. 6. Proposition. If M satisfies (ACC), then every submodule of M can be written as an intersection of a finite number of irreducible submodules of M . Y be the set of all submodules of M which cannot be written as an intersection of a finite number of irreducible submodules of M . If Y is empty, we have finished. Suppose that Y is not empty; then Y has a maximal element N . Since N is not irreducible there are submodules L, and L , of M such that N = L, n L a , N c L,, and N c L,.