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Basically Full Ideals in Local Rings View metadata, citation and similar papers at core.ac.uk brought to you by CORE Journal of Algebra 250, 371–396 (2002) provided by Elsevier - Publisher Connector doi:10.1006/jabr.2001.9099, available online at http://www.idealibrary.com on Basically Full Ideals in Local Rings William J. Heinzer Department of Mathematics, Purdue University, West Lafayette, Indiana 47907 and Louis J. Ratliff Jr. and David E. Rush Department of Mathematics, University of California, Riverside, California 92521 Communicated by Craig Huneke Received September 4, 2001 Let A be a finitely generated module over a (Noetherian) local ring R M.We say that a nonzero submodule B of A is basically full in A if no minimal basis for B can be extended to a minimal basis of any submodule of A properly containing B. We prove that a basically full submodule of A is M-primary, and that the fol- lowing properties of a nonzero M-primary submodule B of A are equivalent: (a) B is basically full in A; (b) B =MBA M; (c) MB is the irredundant intersection of µB irreducible ideals; (d) µC≤µB for eachcover C of B. Moreover, if B ∗ is an M-primary submodule of A, then B = MBA M is the smallest basically full submodule of A containing B and B → B∗ is a semiprime operation on the set of nonzero M-primary submodules B of A. We prove that all nonzero M-primary ideals are closed withrespect to thisoperation if and only if M is principal. In rela- tion to the closure operation B → B∗, we define and study the bf-reductions of an M-primary submodule D of A; that is, the M-primary submodules C of D suchthat C ⊆ D ⊆ C∗.IfGM denotes the form ring of R withrespect to M and G+M its maximal homogeneous ideal, we prove that Mn =Mn∗ for all (resp. for all large) positive integers n if and only if gradeG+M > 0 (resp. gradeM > 0). For a regular local ring R M, we consider the M-primary monomial ideals withrespect to a fixed regular system of parameters and determine necessary and sufficient con- ditions for suchan ideal to be basically full. 2002 Elsevier Science (USA) Key Words: basically full ideal; basis of an ideal; closure operation; cover of an ideal; form ring; injective envelope; integral closure of an ideal; irreducible submod- ule; monomial ideal; Noetherian ring; R-sequence; reduction of an ideal; regular local ring; semiprime operation; socle; superficial element. 371 0021-8693/02 $35.00 2002 Elsevier Science (USA) All rights reserved. 372 heinzer, ratliff jr., and rush 1. INTRODUCTION The investigation of the number of generators of ideals and modules over a local ring has a rich history; [1, 3, 11, 24, 25, 29, 30, 31] are just a few of the many references on this topic. We consider here the ideals of a local ring and the submodules of a finitely generated module over a local ring whichsatisfya maximal property withrespect to extension of a minimal basis. Let A be a finitely generated module over a local ring R M.For a nonzero submodule B of A, Nakayama’s lemma implies that elements b1 bn ∈ B are a minimal basis for B if and only if their images in B/MB are a basis for the vector space B/MB over the field R/M; see, for example [14, (4.1), 12, p. 8, or 11, p. 104]. We say that B is basically full in A if no minimal basis of B can be extended to a minimal basis of any submodule of A that properly contains B. In Section 2 we show in Theorem 2.6 that the basically full submodules of A are M-primary. We give in Theorems 2.4, 2.12, and 2.17 several char- acterizations of the basically full submodules of A. In particular, for an M-primary submodule B of A, we show that B is basically full if and only if B =MBA M if and only if there are µB submodules in an irredun- dant representation of MB as an intersection of irreducible submodules of A if and only if µC≤µB for all covers C of B in A. We show in Theorem 3.3 that one basically full ideal in a local ring yields other basically full ideals in related local rings. In Theorem 4.2 we prove ∗ that the function B → B =MBA M is a closure operation on the set of nonzero M-primary submodules of A. Thus B∗, the basically full closure of B, is the smallest basically full submodule of A containing B. In Section 5 we introduce the concept of a bf-reduction of a nonzero M-primary submodule of A.IfC ⊆ B are nonzero M-primary submodules of A, we call C a bf-reduction of B if B ⊆ C∗, and we say that B is bf- basic if it has no proper bf-reductions. We show that most of the standard properties of ordinary reductions also hold for bf-reductions and that B∗ is the largest submodule C of A withthefollowing property: if D, E are R-submodules of A with B ⊆ D ⊆ E ⊆ C, then each minimal basis of D extends to a minimal basis of E. In Section 6 we consider implications of the fact that the basically full closure B∗ of an open irreducible submodule B is either B or the unique cover B A M of B. In Theorem 7.1, we prove that Mn is basically full for every positive integer n if and only if GradeG+M > 0, where G+M is the maximal homogeneous ideal of the form ring GM. We prove in Theorem 7.2 that Mn is basically full for all large integers n if and only if GradeM > 0. n k In Theorem 7.4, we show that if altitudeR > 0, then M I +0R M basically full ideals 373 is basically full for all open ideals I and for all large integers n and k.In Theorem 7.5, we prove that every nonzero M-primary ideal of R M is basically full if and only if R is a principal ideal ring. In Section 8 we consider M-primary ideals in a regular local ring R M, which are generated by monomials in a fixed regular system of parameters of R. For suchan ideal I, we observe in Proposition 8.2 that the basically full closure I∗ of I is again a monomial ideal. We prove in Corollary 8.4 that to determine whether I is basically full it suffices to show that a minimal monomial basis of I does not extend to a minimal monomial basis of any properly larger monomial ideal of R. Finally, in Section 9, several examples are given which illustrate some of the results of this paper. These examples are all in a regular local ring of altitude two and concern monomial ideals in a fixed regular system of parameters. Throughout the paper we use ⊂ to denote proper containment, and we use µA to denote the number of elements in a minimal basis of a finitely generated module A over a local ring. A general reference for our notation and terminology is [12]. 2. BASIC PROPERTIES OF BASICALLY FULL SUBMODULES Let R M be a (Noetherian) local ring and A a finitely generated R-module. In this section we define basically full submodules of A, show that such modules are M-primary, and give several characterizations of basi- cally full submodules of A. In particular, a submodule B of A is basically full if and only if MB is an irredundant intersection of µB irreducible submodules of A (which is the smallest possible number) if and only if µC≤µB for eachcover C of B. It is also shown that if htM > 0, then each integrally closed M-primary ideal is basically full. Definition 2.1. A nonzero submodule B of a finitely generated R-module A is said to be basically full in A if no minimal basis of B can be extended to a minimal basis of a submodule of A that properly contains B. For an ideal I ⊆ M of R,wesayI is basically full if I is basically full in R. Concerning the terminology “basically full,” when we first began looking at this concept we used “basically maximal.” However, we later changed to “basically full” because of Rees’ terminology “M-full” for the closely related concept defined in (2.2.1). Remark 2.2. (2.2.1) Let R M be a local ring. The concept of an M-full ideal of R is originally due to D. Rees (unpublished). It appears in work of Goto [5] and Watanabe [29–31]. The definition is as follows 374 heinzer, ratliff jr., and rush [29, p. 102]: (1) If the residue field R/M is infinite, an ideal I ⊆ M is M-full if there exists y ∈ M suchthat MIR yR = I. (2) If R/M is not necessarily infinite, let R be a local ring which is faithfully flat over R withinfinite residue field and withMR as the maximal ideal. Then an ideal I of R is called M-full if IR is M-full in the sense of (1). Let M = m m R, let RX X =RX X = R , and 1 n 1 n 1 n MRX1 Xn let Y = m1X1 +···+mnXn. It follows [29] that, in general, an ideal I ⊆ M of R is M-full if and only if MIR R YR = IR .
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