A. V. Roiter

CATEGORIES WITH DIVISIBILITY AND INTEGRAL REPRESENTATIONS

(Presented by Academician P. S. Novikov on 31 V 1963)

1. In this section we shall for the most part adhere to the notation adopted in (¹). For an arbitrary category \(K\) one can construct a category \(\widetilde K\), whose objects are the objects of \(K\), and \(\widetilde H(a,b)\) is the set of all nonempty subsets of \(H(a,b)\); moreover, if \(S \in \widetilde H(a,b)\), \(T \in \widetilde H(b,c)\), then \(ST\) will be understood to be the collection of all mappings of the form \(\sigma\tau\), where \(\sigma \in S\), \(\tau \in T\). We shall regard the category \(K\) as a subcategory of the category \(\widetilde K\), identifying a subset consisting of a single mapping with this mapping. We shall call a pair \((K,L)\) a category with divisibility if \(L\) is a subcategory of the category \(\widetilde K\) containing all objects (i.e., all identity mappings) from \(K\).

Denote by \(L(a,b)\) the set of mappings from \(a\) to \(b\) in the category \(L\) \((L(a,b) \subseteq \widetilde H(a,b))\). We shall say that \(a\) divides \(b\) \((a/b)\) if \(L(a,b)\) is nonempty. The divisibility relation is, obviously, reflexive and transitive, i.e., it is a quasi-ordering relation (²) on the class of objects of the category \(K\). We shall say that the objects \(a\) and \(b\) are associated if \(a/b\) and \(b/a\). The association relation is an equivalence relation, and the divisibility relation induces a partial ordering relation on the collection of classes of associated objects.

In what follows we shall assume that the category \(K\) has zero objects and, consequently, zero mappings. We shall say that \((K,L)\) is a category with nonzero divisibility if, for any two nonzero objects \(a\) and \(b\) of the category \(K\), the zero mapping \(\omega_{ab}\) is not contained in \(L(a,b)\).

Proposition 1. If \((K,L)\) is a category with nonzero divisibility, \(a\) and \(b\) are associated nonzero objects of \(K\), and the semigroup \(H(a,a)\) is finite, then there exist \(\varphi: a \to b\), \(\psi: b \to a\) such that \(\varphi\psi\) is a nonzero idempotent in \(H(a,a)\), and \(\psi\varphi\) is a nonzero idempotent in \(H(b,b)\).

Indeed, the objects \(a\) and \(b\) are associated, hence \(u \in L(a,b)\), \(v \in L(b,a)\), \(uv \in W\). \(W\) is a subsemigroup of the semigroup \(H(a,a)\). Obviously, for every \(n\), \(W^n \ne 0\). But then, as is known (³), there is an \(\alpha \in W\) such that, for every \(n\), \(\alpha^n \ne 0\). Since in a finite semigroup every element in some power is equal to an idempotent, it follows that \(\alpha^n=\beta\), \(\beta^2=\beta \ne 0\), \(\beta=\varphi\psi\), where \(\varphi \in H(a,b)\), \(\psi \in H(b,a)\). Putting \(\psi=\psi\varphi\psi\), we obtain \(\varphi\psi=\beta \in H(a,a)\), \(\psi\varphi=\gamma \in H(b,b)\), \(\gamma^2=\gamma \ne 0\).

Proposition 1 can be somewhat strengthened by replacing the requirement of finiteness of \(H(a,a)\) by certain weaker conditions. In particular, if \(K\) is an additive category, then it suffices to require that the ring \(H(a,a)\) satisfy the maximality and minimality conditions. On the other hand, it is easy to show that Proposition 1 ceases to be true if no finiteness-type restrictions are imposed on \(H(a,a)\).

2. Let now \(K\) be a category of modules over an associative ring with identity. Let \(A,B \in K\), \(A'\) be a submodule of the module \(A\), and \(T\) be a subset

in \(\operatorname{Hom}(A,B)\); denote by \(A'T\) the submodule of the module \(B\) generated by elements of the form \(a't\), where \(a'\in A'\), \(t\in T\). We shall call the set \(T\) epimorphic if \(AT=B\).* Since the product of two epimorphic sets is again an epimorphic set, and a set consisting of a single identity mapping is also epimorphic, we can construct \(L\), denoting by \(L(A,B)\) the set of epimorphic sets lying in \(\operatorname{Hom}(A,B)\), i.e., introduce a divisibility relation on the category of \(K\)-modules. Now \(A/B\) means that \(A\operatorname{Hom}(A,B)=B\).

A decomposition of a module \(A\) into a direct sum \(A_1\oplus\cdots\oplus A_k\) will be called normal if \(A_i\) divides \(A_j\) for \(i<j\). A module that cannot be decomposed into a normal direct sum will be called normally indecomposable.

Proposition 2. Let \(M\) be a commutative Noetherian ring with identity, and let \(U\) be an ideal of the ring \(M\) such that
\[ \bigcap_{k=1}^{\infty} U^k=0, \]
the factor ring \(M/U\) satisfies the minimum condition, and the ring \(M\) is complete as a topological space with the topology induced by the ideals \(U^k\). Let \(K\) be the category of finitely generated \(\Lambda\)-modules, where \(\Lambda\) is an \(M\)-algebra with a finite number of generators. Finally, let \(A\) and \(B\) be modules from \(K\), with \(B\) dividing \(A\) and \(B\) normally indecomposable. Then every exact sequence \(A\to B\to 0\) splits.

Let us note that if \(M\) is a ring satisfying the minimum condition, then the zero ideal may be taken as \(U\). Let us also note that the assertion will hold all the more if \(B\) is indecomposable in the usual sense. The proof of Proposition 2 in the case where \(M\) is a ring satisfying the minimum condition and the module \(B\) is indecomposable coincides exactly with the proof of Proposition 1. In the general case the proof becomes only technically more complicated.

1. Let \(K\) be an arbitrary category. A subobject \((b,\mu)\) of an object \(a\) will be called a supercharacteristic subobject if every \(\nu:b\to a\) can be represented in the form \(\alpha\mu\), where \(\alpha:b\to b\).

In group theory, as is well known, characteristic and fully characteristic subgroups are considered. A supercharacteristic subgroup, i.e. a subgroup that is a supercharacteristic subobject in the sense of the definition given above, is, of course, fully characteristic and, a fortiori, characteristic. However, every proper subgroup of an infinite cyclic group is fully characteristic, but not a supercharacteristic subgroup.

We shall need the concept of a supercharacteristic submodule, for which one may also give the following definition: a submodule \(A'\) of a module \(A\) is called supercharacteristic if \(A'\operatorname{Hom}(A',A)=A'\). The following almost obvious proposition establishes the connection between supercharacteristic submodules and the divisibility relation on the category of modules introduced in the preceding point.

Proposition 3. The module \(A\operatorname{Hom}(A,B)\) is a supercharacteristic submodule of the module \(B\) for every \(A\). The mapping that assigns to each module \(A\) of the category \(K\) the supercharacteristic submodule \(A\operatorname{Hom}(A,B)\) of the module \(B\) induces a homomorphism of the partially ordered set of classes of associated modules of the category \(m\) into the set of supercharacteristic submodules of the module \(B\), partially ordered by inclusion.

From Proposition 2 it follows immediately that

Proposition 4. If \(K\) is a category of modules satisfying the conditions of Proposition 2, then the exact sequence \(A\xrightarrow{\varphi}B\to 0\), where

* Let us note that the notion of an epimorphic, and also of a monomorphic, set of mappings can be naturally defined in an arbitrary category, by analogy with the usual definitions of epimorphism and monomorphism in category theory.

the module \(B\) is normally indecomposable, splits if and only if \(A'\varphi=B\), where \(A'=B\operatorname{Hom}(B,A)\).

1. In this item \(Z\) is the ring of rational integers, \(Z_p\) is the ring of integral \(p\)-adic numbers, \(\Lambda\) is a finitely generated \(Z\)-free \(Z\)-algebra with identity, and \(\Lambda_p\) is a finitely generated \(Z_p\)-free \(Z_p\)-algebra with identity. We shall call the category of finitely generated \(Z\)-(\(Z_p\)-)free \(\Lambda\)-(\(\Lambda_p\)-)modules the category of integral (\(p\)-adic) representations.

First of all, note that in the category of \(p\)-adic representations the conditions, and hence also the assertions, of Propositions 2 and 4 are satisfied. From Proposition 2 and the uniqueness of decomposition into indecomposables in the category of \(p\)-adic representations \((^4)\) it follows:

Proposition 5. In each class of associated modules of the category of \(p\)-adic representations there is one and only one normally indecomposable module.

If \(A\) is a module lying in the category of integral representations, then for every \(p\) the module \(A_p=A\otimes_Z Z_p\) is defined, lying in the category of \(p\)-adic representations of the ring \(\Lambda_p=\Lambda\otimes_Z Z_p\).

Proposition 6. \(A/B\) if and only if \(A_p/B_p\) for all \(p\).

The author does not know whether, in the category of integral representations, the assertions of Propositions 2 and 4 are satisfied. However, it can be proved that

Proposition 7. Let \(K\) be a category of modules satisfying the conditions of Proposition 2, or the category of integral representations. Then the exact sequence \(A \xrightarrow{\varphi} B \to 0\) splits if and only if, for every supercharacteristic submodule \(B'\) of the module \(B\) such that \(B'\operatorname{Hom}(B,B')=B'\), the equality \(A'\varphi=B'\) holds, where \(A'=B'\operatorname{Hom}(B,A)\).

Note that if \(\Lambda\) is a semisimple ring, then, since there exists only a finite number of nonisomorphic modules of a given dimension \((^5)\), every module from the category of integral representations of the ring \(\Lambda\) has only a finite number of distinct supercharacteristic submodules.

From Proposition 7 one may obtain the following corollary: If \(A\) and \(B\) are two modules from the category \(K\) of integral representations, and \(A_p\) is isomorphic to \(B_p\) for every \(p\), then for some \(n\)
\(A^{(n)}=B\oplus X\) and \(B^{(n)}=A\oplus Y\), where \(A^{(n)}\) (\(B^{(n)}\)) is the direct sum of \(n\) copies of the module \(A\) (\(B\)), and \(X\) and \(Y\) are certain modules from \(K\).

Let us also note that everything said about integral and \(p\)-adic representations carries over almost verbatim to representations over Dedekind rings and their completions.

The author expresses his gratitude to his scientific adviser D. K. Faddeev.

Institute of Mathematics
Academy of Sciences of the Ukrainian SSR

Received
16 V 1963

REFERENCES CITED

1. A. G. Kurosh, A. H. Livshits, E. F. Shulgeifer, UMN, 15, 6 (1960).
2. G. Birkhoff, Lattice Theory, Moscow, 1952.
3. N. Jacobson, Structure of Rings, Moscow, 1961, p. 288.
4. Z. I. Borevich, D. K. Faddeev, Vestnik Leningrad Univ., 7 (1959).
5. H. Zassenhaus, Abh. math. Sem. Hansischen Univ., 12 (1938).