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Uniform Dimensions and Subdirect Products

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Uniform Dimensions and Subdirect Products Pacific Journal of Mathematics UNIFORM DIMENSIONS AND SUBDIRECT PRODUCTS JOHN DAUNS Vol. 126, No. 1 November 1987 PACIFIC JOURNAL OF MATHEMATICS Vol. 126, No. 1, 1987 UNIFORM DIMENSIONS AND SUBDIRECT PRODUCTS JOHN DAUNS Over a ring R, £ denotes injective hulls, Z singular submodules. GENERAL THEOREM. For any ring R and any infinite regular cardinal σ, conditions (1), (2), and (3) are all equivalent: (1) Any direct sum of nonsingular right ideals of R contains fewer than σ nonzero summands. (2) // { Wy I γ e Γ} is any indexed set of modules with all ZWy = 0, then σ thesubmodule Π E(JFγ) - {x - (jcγ) e UE(Wy) | (support x\<σ} is infective. (3) For any family {Wy |γ e Γ} having all ZWy = 0, the submodule U°Wy < UWy is a complement. COROLLARY. Every ring R satisfies (1), (2), and (3) for a unique smallest infinite regular cardinal σ = σ(R). THEOREM. For any module M with ZM = 0, E(M) = C θ Z), where C contains no uniform submodules and D = ΪI{E(Dr)|τ GE). The submodules C, D, and the Dτ are all unique. Each DT is a direct sum of isomorphic indecomposable injectives all of the same type T. D The cardinal number of such summands of Dτ is the r-dimension of M. More general uniform dimensions are constructed for arbitrary modules. 1. Introduction. For any unital right i?-module M, its injective hull Λ Λ Λ M is uniquely up to isomorphism of the form M = Z2M θ C θ D, where C is continuous, and D discrete (Theorem 1). If ZM is the singular submodule, then the torsion submodule Z2M of M is defined by Z[M/ZM] = (Z2M)/ZM. Now let M be torsion free, i.e., ZM = 0. A type r is an equivalence class, of nonsingular uniform modules all having isomorphic injective hulls, and Ξ = {T} is the set of all types. Then Mτ = Σ{U\U < M, U e T } < M is an intrinsic submodule of M. The τ-dimension of M is the cardinality of the nonzero summands in any maximal direct sum of submodules of Mτ. More general dimensions Ω — dM and corresponding intrinsic sub- modules MΩ are studied, where Ω is a disjoint union of types. Special important cases are Ω = T a single type, and at the other extreme, Ω = Σ, the class of all torsion free uniform modules. A result of Teply ([Tl; p. 443, Theorem 1.2] and [T2; p. 451, Theorem 2.1]) says that every K0-product (s direct sum) of torsion free injectives is injective if and only if the Goldie dimension of R/(Z2R) is finite. Here the analogue for higher cardinals σ > So is given in Theorem V. Counter- examples 4.3 show that a complete generalization is impossible. 2 JOHN DAUNS In Theorem IV, the hypothesis (1) of the General Theorem II is replaced by the condition (Γ) τ - dR < σ. Even in the finite case when σ = Ko, the results are new. Then it is an immediate corollary of Theorem IV (with Ω = T and σ = So) that the following conditions (1), (2), and (3) are all equivalent: (1) T - d(R/Z2R) is finite. (2) Every direct sum of torsion free indecomposable injective modules of type T is injective. (3) Every direct sum of torsion free uniform modules of type T is a complement submodule in their full direct product. 1. Preliminaries. Standardized terminology, notation, symbols, as well as some facts used throughout the rest of this note are all given at once. 1.1. NOTATION. Modules will be right unital over a ring R. Right i?-submodules are denoted by " < " essential or large ones by " < < " (the same symbol as for submodules—except twice as large); "<" denotes ideals in any ring. In c , D ,<,<<, and < equality is also allowed. However, where important, or for emphasis, sometimes also c , D , and < will be used. Slashes through a symbol denote its negation: <£ , 2 , < < , and £ . In particular "A < < B" means that A < B but that A is not a large submodule of B. If M is a module and 0 Φ Y c M a subset, define Y±= {r <E R\Vy €Ξ 7, yr = 0} = {r\Yr = 0} < R. In particular, M±<R. If m e M and K < M, set m ± = {m} x < R, and for m + K e M/K, define m~ιK = ι (m + A:)J-= {r <= R\mr e #} < i?. Note that for αeϋ, {ma)~K = a~ιm-ιK, but only a{ma)-lK c m"1^. Injective hulls of right i?-modules are denoted by " Λ " as well as Λ "E", e.g. M = EM = E(M), where E will be used when M is given by a formula. 1.2. Singular submodules. The singular and second singular submod- ules of M are ZM = {m <E M\m± < < R} < M and Z[M/ZM] = (Z2M)/(ZM), where (a) Z[M/Z2M] = 0. Also write ZM = Z(M) and Z2M = Z2(M). To avoid confusing a nonsingular module (ZM = 0) with a not singular module (only ZM Φ M), M will be called torsion free if ZM = 0, and torsion if Z2M = M. Although the functor Z does not give a hereditary torsion theory, Z2 does, and Z2M = 0< =» ZM = 0. Furthermore, for M = R, (b) Z2R<R. Since (c) MZ2i? c Z2M, if (d) ZM = 0, then M(Z2R) = 0. UNIFORM DIMENSIONS AND SUBDIRECT PRODUCTS 3 The properties of Z2 used repeatedly throughout here are that (e) ZM < < Z2M and that (f) Z2M < M is the unique smallest complement submodule of M containing ZM. In particular, (g) Z2(EM) = E(ZM) is a direct summand of EM. Also if (h) ZR = 0, then ZM = Z2M. L Λ The symbols < , < < , < < , ~ , ~\ , E, Z, and Z2 always refer to right i?-submodules, and never ever to other rings, e.g. those derived from R such as R/Z2R, or R/ZR, or E(i?/Z2i?). 1.3. Cardinals. The cardinality of any set is denoted by "| |". The successor cardinal of any cardinal whatever such as \X\ is denoted by + \X\ , e.g. N«= tta+ι> An infinite cardinal K is regular if and only if it is not a sum (i.e. a disjoint union) of the form K = Σ{κι \i e /} = U{κ |/ G /}, where |/| < K as well as κέ < K for all i e /. (See [HJ; p. 161-163; 3.1, 3.2].) 1.4. A sum of submodules of some bigger module is denoted by "Σ", whereas both an internal and an external direct sum will be written as "φ". For modules {Wγ\yf=T}, form φ {Wy |γ e Γ} = φ Wy < Π{PFγ |γ e Γ} = TlWy. Whenever an index set is omitted, not only as above in φ and Π, but also in Σ, U, or Π, it will then be understood that the index ranges over the largest possible index set, and never over some proper subset. For x = {xy\y e Γ} = (xγ)γeΓ = (xy) e ILWy, and for 7 c HWy, define their supports to be the subsets suppx = {γ e Γ | xy Φ 0} and SuppΓ = U{s\xρpy\y e 7} of Γ. Thus Supρ{x) = suppx. 1.5. DEFINITION. For an infinite cardinal σ, the σ-product of an indexed set of sets Wy, γ e Γ, modules, or rings is defined to be the σ σ subdirect product Π { Wy |γ e Γ} = {x e UWy ||suppx| < σ} = U Wy. σ Thus for σ = No = |ω| = |ωo|, Π is just the ordinary direct sum. 1.6. For an infinite cardinal σ and modules Wy, γ e Γ, and one single element x == (jcγ) e UWy, define Ai < R, i G / as follows: / = {/ c Γ| |/| < σ}, At = Πix^ |γ «£ /}. Note that also^z = Πfx^ |γ e suppx; γί/}. Hence Suppx^ί^ c /, and At is the unique largest right ideal of R with this property. If ίcje /, then ^4 c Aj. If /, K Q I are any subsets, then Π{^l^|/:^A'}=^nA:, wΛ^r^ Πί: = n{A:|A:Gϋ:}G/, whereas Σ{Aj,\ j e /} c ^u/? provided that U/ σ σ e /, i.e. that |U/| < σ. Consequently, for x + U Wy e UWy/U Wy, σ ± 1 σ (x + Π Wy) = χ- π Wy 4 JOHN DAUNS 2. Types and Uniform Dimensions. First it is shown that any injective module M can be decomposed as a direct sum M = C Θ C" Θ Df θ D of four different kinds of modules; C θ C" is continuous or uniform free; Z)' θ D is discrete, that is contains an essential direct sum of uniform submodules; C θ Df is torsion, while Cθΰis torsion free. 2.1. LEMMA. Every injective module M is uniquely of the form M = F Φ G up to isomorphism, where G contains an essential direct sum of uniform modules and F is uniform free. Proof. Zorn's lemma applied to the set of sets of uniform submodules whose sum is direct produces M = F θ G. Similarly, let also M = Fx θ Gv By the common refinement property ([W; p. 268, Cor. 1.1]), decom- pose further F = F' θ F", G = G' Φ G", etc. so that M = F θ F" θ G' Φ G" = F/ θ F/' θ G( Φ Gί', where Fr = F{; G" = G['\ F" s G(; and Gr s F/'. r/ r Thus F = G( = 0 and G = F/' = 0. Hence F = F' = F{ = Fl9 and r G = G" s G( = Gx.
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