A Class of Divisible Modules
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Pacific Journal of Mathematics A CLASS OF DIVISIBLE MODULES MARK LAWRENCE TEPLY Vol. 45, No. 2 October 1973 PACIFIC JOURNAL OF MATHEMATICS Vol. 45, No. 2, 1973 A CLASS OF DIVISIBLE MODULES MARK L. TEPLY The ^^divisible jβ-modules are defined in terms of a hereditary torsion theory of modules over an associative ring R with identity element. In the special case where J^~ is the usual torsion class of modules over a commutative in- tegral domain, the class of J^divisible modules is precisely the class of divisible modules M such that every nonzero homomorphic image of M has a nonzero /^-divisible submodule. In general, if J7~ is a stable hereditary torsion class, the class of ^-divisible modules satisfies many of the traditional properties of divisible modules over a commutative integral domain. This is especially true when J^~ is Goldie's torsion class 2^. For suitable ^\ the splitting of all ^divisible modules is equivalent to h.d.Q^- ^ 1, where Q^ is the ring of quotients naturally associated with ^T Generalizations of Dedekind domains are studied in terms of ^^divisibility. 1* Notation, terminology, and preliminary results* In this paper, all rings R are associative rings with identity element, and all modules are unitary left iϋ-modules. B^f denotes the category of all left i?-modules. E(M) denotes the injective envelope of Me R<^fί. In homological expressions, the "R" will be omitted for convenience 1 in printing (e.g. Ext^ = Ext and h.d RQ — h.d. Q). Following S.E. Dickson [4], we call a nonempty subclass J7~ of R^/^ a torsion class if ^ is closed under factors, extensions, and arbitrary direct sums. ^7~ is called hereditary if it is closed under submodules. Modules in J7~ are called torsion. Every torsion class determines in every A e R^ a unique maximal torsion submodule ^~{A) is called the torsion submodule of A, and J7~{A\j?~{A)) = 0. Modules in ^~ = {Ae R^t | J7~(A) = 0} are called torsίonfree, and the torsionfree class Jf is closed under submodules, extensions, and direct products. ^ is hereditary if and only if j^~ is closed under injective envelopes. Throughout this paper, ^~ will denote a hereditary torsion class will denote the torsionfree class corresponding to j?~\ hence is a hereditary torsion theory [4], [10], and [17]. Each such is uniquely associated with a topologizing and idempotent filter F(^r) - {I \ JG R and R/IeJ^} of left ideals of R. In [5] the right derived functors for a hereditary torsion class J7~ are examined. These derived functors are given by - ^(A), E'jr (A) - 653 654 MARK L. TEPLY and R%(A) = Rn^\E{A)\A) for n ^ 2. If A is injective, then R^(A) = 0 for Λ ^ 1. If ie/; then BιAA) = (^~E(A)/A). A module M is called ^-injective ([8], [9], and [15]) if Ext1 (Γ, Af) = 0 for all Γey; (Note: in [14] ^injective modules are called i?7(^")-divisible.) This happens exactly when ^(E(M)/M) = 0 (see [8]). Hence a module A e ^ is ^injective if and only if RAA) = 0. If 0 > A >B >C >0 is any short exact sequence, then so is the induced sequence 0 JΓ(B) -> JT(C) -> B\r(A\ - RιAB) - RAQ -> R2AΛ) - . A torsion class ^~ is called stable if ^~ is closed under injective envelopes. R'AT) = 0 for all Te_^l and if ^" is stable, then RAT) = 0 for all Γe^ and for all n ^ 1. This and the long exact sequence in the preceding paragraph imply that if ^~ is stable and B e R^fί, then RAB) = 0 if and only if RAB/^~(B)) - 0. The submodule EAM) of jE'(ikf) is the (unique) largest module satisfying MS E^(M) S #(ikf) and EAM)/Me^. We call JS7^(Λf) the ^-injective envelope of Λf. E^(M) exists [14] and is .^injective. If iϋe^7 then we will use Q^- to denote EAR)* Q^ has a ring structure [8] and has received attention in a large number of papers (e.g.[7], [8], [9], and [15]). A module M is said to split if J7~(M) is a direct summand of M. We now state a lemma giving some properties of ^injective modules. LEMMA 1.1. (1) S~ is stable if and only if every ^-injective modules splits. (2) A direct summand of a Jf~-injective module is ^"-injective. (3) If A is a ^-injective module and if θ: A—>B is an epi- morphism, then θ~ι{^~(B)) = {x e A \ θ(x) e ^(B)} is ^-injeetive. (4) If j^Γ is stable, then every ^-injective module in j^7~ is infective. Proof. (1) If ^ is stable and A is ^injective, then E(A)/A so JT(E(A)) S A. By [2, Prop. 2.1], E(A) - ^~(E(A)) 0 F for some Fe^. Hence A = J^(#(A)) © (F n A). Since J^(#(A)) - ^~(A) in this case, then A splits. The converse is immediate from [2, Prop. 2.1]. (2) This is well-known and straight forward to prove. (3) Note that - A g(A) _ B A CLASS OF DIVISIBLE MODULES 655 So by the definition of ^injective envelope, E^(θ~ι(JΓ~(B))) = (4) Let Tej^~ be j^injective; so E(T)e^ and E(T)/TejK Since j?~ is closed under homomorphic images, E(T)/T e J7~ Π ^ = {0}; thus E{T) = Γ. We now turn our attention to an important special torsion theory. The class & of Goldie torsion modules is the smallest class contain- ing all isomorphic copies of all factor modules A/B, where B is an essential submodule of A (see [1], [2], [16], and [17]). The correspond- ing Goldie torsionfree class Λ^ is exactly the class of nonsingular modules. Sf is hereditary and stable; if R e <ylr, then & is precisely the class of singular modules. So if R is a commutative integral domain, then %? and <yίr coincide with the usual torsion and torsion- free classes, respectively. This makes (g^, Λ^) a "natural" torsion theory to consider when one is trying to generalize the usual results about torsion modules over an integral domain. Moreover, the &- injective modules are just the injective modules; and if R e ^V, then Q, ~ E(R) and the filter F(&) is the set of essential left ideals of R. 2* .^divisible modules* Historically the class of divisible modules has been very useful in studying modules over a commuta- tive integral domain. Frequently subclasses of the class of divisible modules have also yielded interesting results (e.g. see [11] and [12]). One such subclass studied by Matlis is the class of h-divisible modules, i.e., those modules which are homomorphic images of injective modules. A class related to the Λ-divisible modules is the class of divisible modules M such that every nonzero homomorphic image of M has a nonzero A-divisible submodule. For reasons indicated later, we refer to this latter class as the ^/-divisible modules. We shall show that not only does the class of ^-divisible modules satisfy many interest- ing properties but also it allows us to "smooth out" some of Matlis' results. For example, Matlis proves [11, Cor. 2.6] that every divisible module splits if and only if (1) h.d. Q ^ 1 (where Q is the quotient field of R) and (2) T = Ext1 (Q/R, T) for every torsion divisible module T with 0 as its only ^-divisible submodule. We remove condition (2) by only considering ^-divisible modules: every ^-divisible module splits if and only if h.d. Q <^ 1 (see Corollary 4.6). However, we will not limit our investigation to modules over a commutative integral domain. We are able to state our entire theory in terms of hereditary torsion theories (^, JF*) of modules over an associative ring R with identity element. To do this, we begin by defining subclasses ^a of R^fS for each ordinal number a. 656 MARK L. TEPLY ^ = {M e R^£ I M is a homomorphic image of a direct sum of ^^injective modules}. If a is not a limit ordinal, then <g^ = {Me ^ I IN S Λί" such that iVe ^_t and M/ΛΓe <gy. If a is a limit ordinal, then <g^ = {Λf e R^€ \ M = U JJ< JVjs where JV), 6 ^ for /S < a}. This transίinite definition is similar to the construction of a Loewy series [6]. Clearly <g^ S ^α for β ^a. It is straight forward to show that each <g^ is closed under homomorphic images and direct sums. If R is a commutative integral domain and if the torsion class is taken to be ^, it follows that each ^ is a subclass of the (usual) class of divisible modules. Also, over a commutative integral domain, any divisible module in ^ is injective. These two facts, plus the fact that each ^ depends on ^7 motivates the terminology, "^divisible module", for any hereditary torsion theory (_^7 ^) over anY ring. DEFINITION. A module M is called ^^divisible if and only if (i) M) is .^injective and (ii) ^~(M) e ^a for some ordinal a. will denote the class of all ^divisible modules. EXAMPLE 2.1. If R is a commutative integral domain, then a module M is ^-divisible if and only if (a) M is divisible and (b) every nonzero homomorphic image of M has a nonzero A-divisible submodule. Let MeQί^. From the construction of the classes ^, the closure properties of divisible modules, and the definition of gf-divisi- ble, it follows that M is divisible. Let θ: M—> X be an epimorphism. If ker θ does not contain gf (ikf), then it follows from (ii) that (&(M) + ker#)/ker θ contains an /^-divisible submodule.