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Monoids and Categories of Noetherian Modules UNIVERSITY OF CALIFORNIA Santa Barbara Monoids and Categories of Noetherian Modules A Dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics by Gary John Brookfield Committee in Charge: Professor K. R. Goodearl, Chairperson Professor B. Huisgen-Zimmermann Professor J. Zelmanowitz June 1997 ii The Dissertation of Gary John Brookfield is approved Committee Chairperson June 1997 iii VITA August 3, 1954 — Born — Southampton, UK 1978 — B.Sc. Physics, University of British Columbia 1981 — M.Sc. Physics, University of British Columbia PUBLICATIONS Direct Sum Cancellation of Noetherian Modules, accepted by the Journal of Algebra, March 1997 FIELDS OF STUDY Major Field. Mathematics Specialties: Ring Theory, Noetherian Modules Professor K. R. Goodearl iv Abstract Monoids and Categories of Noetherian Modules by Gary John Brookfield In this dissertation we will investigate the structure of module categories by considering how modules can be constructed by extensions from other modules. The natural way to do this is to define for a module category S, a commutative monoid (M(S), +) and a map Λ: S → M(S) so that the relevant information about S is transferred to M(S) via Λ. Specifically, we construct M(S) so that if A, B, C ∈ S and there is a short exact sequence 0 → A → B → C → 0, then Λ(B) = Λ(A) + Λ(C) in M(S). We also require that M(S) be the largest monoid with this property, meaning that M(S) has the universal property for such maps. Let R be a ring, R-Mod the category of left R-modules and R-Noeth the category of Noetherian left R-modules. Then we will prove • M(R-Mod) is a conical commutative refinement monoid. • M(R-Noeth) is semi-Artinian and strongly separative, meaning that 2a = a + b =⇒ a = b for all a, b ∈ M(R-Noeth). • The Krull dimension of a Noetherian module can be determined from its image in M(R-Noeth). • If R is left fully bounded Noetherian, in particular, if R is commutative Noetherian, then M(R-Noeth) is Artinian, primely generated and weakly cancellative, and has ≤-multiplicative cancellation. The strong separativity of M(R-Noeth) leads to one of the most interesting results of this dissertation: If A, B, C are left R-modules such that A⊕C =∼ B ⊕C with C Noetherian, then A and B have isomorphic submodule series. v Contents A. Introduction 1. Introduction . 1 B. Ordered Classes 2. Artinian Ordered Classes . 11 3. The Length Function on Ordered Classes . 25 4. The Krull Length of Artinian and Noetherian Modules. 37 C. Commutative Monoids 5. Commutative Monoids. 41 6. Order and Monoids . 51 7. Refinement and Decomposition Monoids . 59 8. Cancellation and Separativity. 67 9. Weak Cancellation and Midseparativity . 76 10. Groups and Monoids . 87 11. Primely Generated Refinement Monoids . 94 12. Artinian Decomposition Monoids . 109 13. Artinian Refinement Monoids . 115 14. Semi-Artinian Decomposition Monoids . 119 15. Semi-Artinian Refinement Monoids . 132 D. Modules 16. Monoids from Modules . 141 17. Noetherian and Artinian Modules. 149 18. The Radann Map. 156 19. Noetherian and FBN Rings . 160 References . 171 1 1 Introduction It is the hope of all module theorists that modules can be understood and classified by considering how they can be built from a family of simpler building blocks. The prototype for this idea is the classification of Noetherian Z-modules, that is, finitely generated Abelian groups: Every such group is a finite direct sum of the groups Z and Zpn for p, n ∈ N and p prime. Thus the building blocks for finitely generated Abelian groups are the groups Z and Zpn , and the glue that holds them together is the direct sum operation. Moreover, two finitely generated Abelian groups are isomorphic if and only if they are built from the same building blocks. This is the best possible situation and reflects a complete understanding of Noetherian Z-modules. This happy situation, when it occurs, has many simple consequences. Some of these which will be important in discussing other module categories, we list here: • Cancellation: If A, B and C are Noetherian Z-modules such that A ⊕ C =∼ B ⊕ C, then A =∼ B. • Multiplicative Cancellation: If A and B are Noetherian Z-modules such that An =∼ Bn for some n ∈ N, then A =∼ B. ∼ • Refinement: If A1, A2, B1 and B2 are Noetherian Z-modules such that A1 ⊕ A2 = B1 ⊕ B2, then there are Noetherian Z-modules C11, C12, C21 and C22 such that ∼ ∼ A1 = C11 ⊕ C12 A2 = C21 ⊕ C22 ∼ ∼ B1 = C11 ⊕ C21 B2 = C12 ⊕ C22. If we write A . B when A is isomorphic to a direct summand of B, then we have • Antisymmetry: If A and B are Noetherian Z-modules such that A . B . A, then A =∼ B. • .-Cancellation: If A, B and C are Noetherian Z-modules such that A ⊕ C . B ⊕ C, then A . B. • .-Multiplicative Cancellation: If A and B are Noetherian Z-modules such that n n A . B for some n ∈ N, then A . B. • Decomposition: If A, B1 and B2 are Noetherian Z-modules such that A . B1 ⊕ B2, then there are Noetherian Z-modules C1, C2 such that C1 . B1, C2 . B2 and ∼ A = C1 ⊕ C2. • Descending Chain Condition: If A1 & A2 & A3 & ... is a decreasing sequence of ∼ Noetherian Z-modules then there is an N ∈ N such that An = AN for all n ≥ N. All of the above properties derive from the classification theorem for Z-Noeth. (The notation used in this dissertation is explained at the end of this section.) The reason for their importance is that, for other rings and in other circumstances, they occur even without a complete classification theorem, and hence can be considered consolation prizes when a complete classification is not possible. Indeed, almost no category of modules can be classified the simple way that Z-Noeth is. Even though the full classification may not be possible, there is still some hope that some of these other weaker properties may occur. Section 1: Introduction 2 Before considering more general module categories, we note the special properties that the building blocks of Z-Noeth have: A nonzero Noetherian Z-module P is one of the building blocks described above if it has the following property: • Primeness: If A and B are Noetherian Z-modules such that P . A ⊕ B, then either P . A or P . B. It will be very useful to add an element of abstraction to the classification of Noetherian Z-modules which will enable us to discuss the successes and failures of similar classification schemes for other module categories: Definition 1.1. Let R be a unital ring and S ⊆ R-Mod a class of left R-modules which is closed under finite direct sums and contains the zero module. For any module A ∈ S, let {=∼ A} be its isomorphism class. Let V (S) be class of all isomorphism classes of S. We will write + for the operation on V (S) induced by the direct sum, that is, {=∼ A} + {=∼ B} = {=∼ A ⊕ B} for all A, B ∈ S. It is easy to see that V (S) is a well defined commutative monoid (though perhaps not a set; see 5.1). The identity element of V (S) is the image of the zero module, {=∼ 0} = {0}, which we will write as 0 ∈ V (S). The classification of Noetherian Z-modules discussed above is then a statement about V (Z-Noeth): ∼ ∼ • V (Z-Noeth) is a free commutative monoid with basis {= Z} and {= Zpn } for p, n ∈ N and p prime (5.13). Corresponding to the properties of Z-Noeth described above are the following monoid properties. These we define for an arbitrary commutative monoid M: P1: Cancellation: If a, b, c ∈ M such that a + c = b + c, then a = b. P2: Multiplicative Cancellation: If a, b ∈ M such that na = nb for some n ∈ N, then a = b. P3: Refinement: If a1, a2, b1, b2 ∈ M such that a1 + a2 = b1 + b2, then there are c11, c12, c21, c22 ∈ M such that a1 = c11 + c12 a2 = c21 + c22 b1 = c11 + c21 b2 = c12 + c22. For A, B ∈ Z-Noeth we have ∼ ∼ ∼ ∼ (A . B) ⇐⇒ (∃C such that A ⊕ C = B) ⇐⇒ (∃C such that {= A} + {= C} = {= B}). Accordingly, in an arbitrary commutative monoid M, we define a relation ≤ by a ≤ b ⇐⇒ ∃c ∈ M such that a + c = b ∼ ∼ for a, b ∈ M (6.1). Thus, A . B in Z-Noeth if and only if {= A} ≤ {= B} in V (Z-Noeth). Corresponding to the remaining properties of Z-Noeth we then get the following monoid properties: P4: Antisymmetry: If a, b ∈ M such that a ≤ b ≤ a, then a = b. P5: ≤-Cancellation: If a, b, c ∈ M such that a + c ≤ b + c, then a ≤ b. P6: ≤-Multiplicative Cancellation: If a, b ∈ M such that na ≤ nb for some n ∈ N, then a ≤ b. Section 1: Introduction 3 P7: Decomposition: If a, b1, b2 ∈ M such that a ≤ b1 + b2, then there are c1, c2 ∈ M such that c1 ≤ b1, c2 ≤ b2 and a = c1 + c2. P8: Descending Chain Condition: If a1 ≥ a2 ≥ a3 ≥ ... is a decreasing sequence in M, then there is an N ∈ N such that an ≥ aN for all n ≥ N. Note that we require an ≥ aN rather than an = aN . See Section 2 for details. The basis elements of V (Z-Noeth) can be characterized within M = V (Z-Noeth) as those elements 0 6= q ∈ M with the following property: • Primeness: If a, b ∈ M such that q ≤ a + b, then either q ≤ a or q ≤ b. Elements of a commutative monoid which satisfy this condition are called prime ele- ments.
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