aa r X i v : . [ m a t h . G R ] F e b SELF-SMALL PRODUCTS OF ABELIAN GROUPS
JOSEF DVOˇR ´AK AND JAN ˇZEMLIˇCKA
Abstract.
For abelian groups
A, B , A is called B -small if the covariant func-tor Hom ( A, −) commutes with all direct sums B ( κ ) and A is self-small providedit is A -small. The paper characterizes self-small products applying developedclosure properties of the classes of relatively small groups. As a consequence,self-small products of finitely generated abelian groups are described. Research of modules whose covariant functor Hom ( M, −) commutes with alldirect sums, which is a condition providing a categorial generalization the notionof finitely generated module, started in 60’s by the work of Hyman Bass [6, p.54]and Rudolf Rentschler [20]. Such modules have appeared as a useful tool in diversecontexts and under various terms (small, Σ-compact, U-compact, dually slender)in ring theory, module theory and in the study of abelian groups. David M. Arnoldand Charles E. Murley published their influential paper [5], which is dedicatedto a particular case of the studied condition by narrowing it to commuting withdirect sums of the tested module itself, in 1974. Groups and modules satisfyingthis restricted condition are usually called self-small in literature. Many interestingresults concerning self-small modules over unital rings in general have appeared later[1, 10, 11, 16, 18], self-small abelian groups proving to be a particularly successfultool [2, 3, 4, 7, 8, 9].The aim of this paper is to deepen the present knowledge about structure ofself-small groups and about possibilities of testing abelian groups for self-smallnessby adopting some ideas of the papers [2, 13, 17] and extending several resultsof [12, 21]. Namely, we deal with the notion of a relatively small abelian group(defined in [2, 16], cf. also relatively compact objects in [17]) which serves as a toolfor characterization of those products of groups that are self-small.Throughout the paper module means a right module over an associative ring withunit and an abelian group is a module over the ring of integers. Note that we willuse the term group instead of abelian group frequently, as non-abelian groups arenot considered here. If A and B are two abelian groups, then Hom ( A, B ) denotesthe abelian group of homomorphisms A → B . A family of groups means a discretediagram in the category of abelian groups, so a family may contain more that onecopy of a group. The set of all prime numbers is denoted by P and we identifycardinals with least ordinals of given cardinality.For non-explained terminology we refer to [14, 15]. Date : February 24, 2021.2000
Mathematics Subject Classification.
Key words and phrases. self-small abelian group, slender group.This work is part of the project SVV-2020-260589. Relatively small groups
Let A , B be abelian groups and N a family of abelian groups. It is well-known(and easy to see) that the functor Hom ( A, −) induces an injective homomorphismof abelian groups Ψ N ∶ ⊕ N ∈N Hom ( A, N ) →
Hom ( A, ⊕ N ) by the rule Ψ N (( f N ) N ) = ∑ N f N (cf. e.g. [17, Lemma 1.3]). Suppose, then, that C is a class of groups and B is an abelian group. We say that A is C -small if Ψ N is anisomorphism for each subfamily N of class C and A is said to be B -small providedit is a { B } -small group (cf. [2, 13, 16, 17]). It is clear that A -small abelian groups A are exactly self-small ones as defined in [5]. Example 1.1. (1) Every finitely generated abelian group is small, so B -small forevery group B . In, particular each finite group is self-small.(2) Let A and B be two abelian groups such that Hom ( A, B ) =
0. Then it iseasy to see that A is B -small.In particular, if p, q ∈ P are different primes, A p is an abelian p -group and A q isan abelian q -group, then A p is A q -small and Z -small. Example 1.2.
It is clear, Q and Q / Z are Q -small groups but neither Q nor Q / Z is Q / Z -small. Furthermore, Q -small groups are precisely groups of finite torsion-freerank by [2, Corollary 4.3.].We start with an elementary observation which translates the definition of arelative small group to an easily tested condition (cf. [20, Section 1], [17, Lemma1.4(2)] and [13, Theorem 1.6(2)]): Lemma 1.3.
Let A and B be abelian groups and C a class of abelian groups.Then A is C -small if and only if for each family N of groups contained in theclass C and every f ∈ Hom ( A, ⊕ N ) there exists a finite family F ⊆ N such that f ( A ) ⊆ ⊕ F . In particular, A is B -small if and only if for every index set I andevery f ∈ Hom ( A, B ( I ) ) there exists a finite subset F ⊆ I such that f ( A ) ⊆ B ( F ) ) .Proof. The argument of the proof is well known; if Ψ N is onto and f ∈ Hom ( A, ⊕ N ) ,then there exist finitely many f i ∈ Hom ( A, N i ) , i = , . . . , n such that Ψ N (⊕ i f i ) = f ,hence f ( A ) ⊆ ⊕ ni = N i . On the other hand, if f ( A ) ⊆ ⊕ ni = N i ⊆ ⊕ N , thenΨ N (⊕ i π N i f ) = f , where π N i denotes the projection onto i -th component. (cid:3) The observation that the concept of relatively small groups is general enough ifwe consider relative smallness over a set of groups (cf. general [13, Lemma 2.1])presents a first application of the previous lemma. To that end, for a class of groupsdefineAdd (C) = { A ∣ A is a direct sumand of ⊕ α < κ C α for some cardinal κ and C α ∈ C} and by Add ( A ) denote Add ({ A }) . Lemma 1.4.
Let A be an abelian group and C be a set of abelian groups. Then thefollowing conditions are equivalent: (1) A is ⊕ C -small, (2) A is C -small, (3) A is Add (⊕ C) -small.
ELF-SMALL PRODUCTS OF ABELIAN GROUPS 3
Proof. (1) ⇒ (3) Put B = ⊕ C , let N be a family of groups contained in Add ( B ) ,and f ∈ Hom ( A, ⊕ N ) . Then for each N ∈ N there exists a cardinal κ N forwhich N ⊆ B ( κ N ) ( N is also a direct summand of B ( κ N ) ), and so f ( A ) ⊆ ⊕ N ⊆ ⊕ N ∈N B ( κ N ) . Since A is B -compact, there exists finite family F ⊆ N such that f ( A ) ⊆ ⊕ N ∈F B ( κ N ) which implies that f ( A ) ⊆ ⊕ F .(3) ⇒ (2) It is obvious since C ⊆ Add (⊕ C) .(2) ⇒ (1) As any group B ∈ C is a direct summand of ⊕ C , the same argument asin the implication (1) ⇒ (3) proves the assertion. (cid:3) Since Add ( B ) = Add ( B ( κ ) ) for an arbitrary group B and a nonzero cardinal κ ,we obtain the following useful criterion: Corollary 1.5.
Let A and B be abelian groups and κ a nonzero cardinal. Then A is B -small if and only if A is B ( κ ) -small. As a consequence, we can formulate a well-known closure property of the classof all self-small groups.
Corollary 1.6.
Let κ be a cardinal and A an abelian group. Then A ( κ ) is self-smallif and only if A is self-small and κ is finite. Let us formulate a variant of the assertion [2, Theorem 4.1.], which generalize theclassical criterion of self-small groups [5, Proposition 1.1] for the case of relativelysmall groups (cf. [13, Lemma 3.3]). Recall that the family ( A i ∣ i < ω ) is said to be ω -filtration of a group A , if it is a chain of subgroups of A , i.e. A i ⊆ A i + for each i < ω , with A = ⋃ n < ω A n . Proposition 1.7.
The following conditions are equivalent for abelian groups A and B : (1) A is not B -small, (2) there exists a homomorphism f ∈ Hom ( A, B ( ω ) ) such that f ( A ) ⊈ B ( n ) forall n < ω , (3) there exists an ω -filtration ( A i ∣ i < ω ) of A such that for each n < ω thereexists a nonzero f n ∈ Hom ( A, B ) satisfying f n ( A n ) = , (4) there exists an ω -filtration ( A i ∣ i < ω ) of A such that Hom ( A / A n , B ) ≠ for each n < ω .Proof. The proof works using similar arguments as in [5, Proposition 1.1].(1) ⇒ (2) By Lemma 1.3 there exists a set I and g ∈ Hom ( A, B ( I ) ) such that g ( A ) ⊈ B ( F ) for any finite F ⊂ I . Then we can construct by induction a sequenceof finite sets I n ⊂ I such that I = ∅ , ∣ I n ∖ I n − ∣ = π I n − g ⊋ ker π I n g forall n < ω where π I n ∈ Hom ( B ( I ) , B ( I n ) ) denotes the natural projection. If we put I ω = ⋃ i < ω I i , then π I ω g ∈ Hom ( A, B ( I ω ) ) represents the desired homomorphism.(2) ⇒ (3) Let f ∈ Hom ( A, B ( ω ) ) satisfy the condition (2) and define A n = f − ( B ( n,ω ) ) where B ( n,ω ) = { b ∈ B ( ω ) ∣ π i ( b ) = ∀ i ≤ n } for natural projections π i ∶ B ( ω ) → B onto the i -th coordinate. Then A = ⋃ i < ω A i and for each i < ω there exist n i > i such that f i = π n i f ≠ f i ( A i ) = ⇒ (4) It is enough to observe that any nonzero f n ∈ Hom ( A, B ) satisfying f n ( A n ) = π ∶ A → A / A n , i.e.there exists nonzero ˜ f n ∈ Hom ( A / A n , B ) for which ˜ f n π = f .(4) ⇒ (1) Let f i ∈ Hom ( A / A i , B ) denote a nonzero homomorphism and define ahomomorphism f ∈ Hom ( A, B ω ) by the rule π i ( f ( a )) = f i ( π A i ( a )) = f i ( a + A i ) for JOSEF DVOˇR´AK AND JAN ˇZEMLIˇCKA each a ∈ A and i < ω . Then f ( A ) ⊆ B ( ω ) since for each a ∈ A there exists n suchthat a ∈ A i for all i ≥ n , hence f ∈ Hom ( A, B ( ω ) ) . On the other hand, f ( A ) ⊈ B ( n ) for any n < ω as π n f ≠ i < ω . Thus A is not B -small by Lemma 1.3. (cid:3) The previous assertion applied on A = B allows us to reformulate [12, Proposition9]. Corollary 1.8.
The following conditions are equivalent for an abelian groups A : (1) A is not self-small, (2) there exists an ω -filtration ( A i ∣ i < ω ) of A such that Hom ( A / A n , A ) ≠ for each n < ω , (3) there exists an ω -filtration ( A i ∣ i < ω ) of A such that for each n < ω thereexists a nonzero ϕ n ∈ End ( A ) satisfying ϕ n ( A n ) = . Example 1.9.
Put P = ∏ p ∈ P Z p . Then Hom ( Q , P ) = Q is P -small. On the other hand, if we put B = P / ⊕ p ∈ P Z p , then there existsexists an ω -filtration ( B i ∣ i < ω ) of B such that Hom ( B / B n , Q ) ≠ n by[12, Example 3]. If we take preimages A n of all B n in canonical projection P → P / ⊕ p ∈ P Z p , then ( A i ∣ i < ω ) forms an ω -filtration of A satisfying Hom ( A / A n , Q ) ≅ Hom ( B / B n , Q ) ≠
0, hence P is not Q -small by Proposition 1.3 (equivalently, wecould use [2, Corollary 4.3.]).2. Closure properties of relative smallness
First, let us formulate several elementary relations between classes of relativelysmall groups.
Lemma 2.1.
Let A , B and C be abelian groups and I be a set. Suppose that A is B -small. (1) If C is a subgroup of A , then A / C is B -small. (2) If C is embeddable into B I , then A is C -small.Proof. (1) Proving indirectly, we assume that A = A / C is not B -small. Then thereexists an ω -filtration ( A i ∣ i < ω ) of A for which Hom ( A / A n , B ) ≠ n < ω byProposition 1.7. If we lift all the groups of the ω -filtration of A to the ω -filtration ( A i ∣ i < ω ) of A satisfying the conditions C ≤ A n and A n / C = A n for each n , thenHom ( A / A n , B ) ≅ Hom ( A / A n , B ) ≠
0, hence A is not B -small by Proposition 1.7.(2) We may suppose w.l.o.g. that C ≤ B I . Assume A is not C -small andconsider the ω -filtration ( A i ∣ i < ω ) of A for which Hom ( A / A n , C ) ≠ ( A / A n , B I ) ≠ n < ω and since foreach nonzero f n ∈ Hom ( A / A n , B I ) there exists i ∈ I such that π i f n ≠
0, we concludethat Hom ( A / A n , B ) ≠ n < ω , a contradiction. (cid:3) Proposition 2.2.
Let A be a self-small abelian group. (1) If f ∈ Hom ( A, A I ) for an index set I , then f ( A ) is self-small. (2) If I ⊆ End ( A ) , then A / ⋂{ ker ι ∣ ι ∈ I } is self-small.Proof. (1) Since A is A -small, f ( A ) is A -small by Lemma 2.1(1). Thus f ( A ) is f ( A ) -small by Lemma 2.1(2).(2) If ϕ ∶ A → A I is defined by the rule π ι ϕ = ι for each ι ∈ I , then ker ϕ = ⋂{ ker ι ∣ ι ∈ I } , hence A / ⋂{ ker ι ∣ ι ∈ I } ≅ f ( A ) is self-small by (1) (cf. also [13, Example2.10]). (cid:3) ELF-SMALL PRODUCTS OF ABELIAN GROUPS 5
The next assertion describes closure properties concerning extensions.
Proposition 2.3.
Let A and C be abelian groups and B ≤ C . (1) If both B and C / B are A -small, then C is A -small. (2) If A is B -small and C / B -small, then A is C -small.Proof. Similarly as in Lemma 2.1, we will use throughout the whole proof thecorrespondence of relative nonsmallness and properties of ω -filtrations given byProposition 1.7.(1) Suppose that ( C n ∣ n < ω ) is an ω -filtration of C . Then ( C n ∩ B ∣ n < ω ) is an ω -filtration of B and ( C n + B / B ∣ n < ω ) is an ω -filtration of C / B . Since B and C / B are A -small, there exists n such that f ( B ) = f ∈ Hom ( C, A ) satisfies f ( B ∩ C n ) =
0, and ˜ f ( C / B ) = f ∈ Hom ( C / B, A ) satisfies˜ f ( C n + B / B ) = f ∈ Hom ( C, A ) such that f ( C n ) =
0, then f ( B ) = f ( C n ∩ B ) = f ∈ Hom ( C / B, A ) for which ˜ f π B = f . Now, ˜ f ( C / B ) = f ( C n + B / B ) =
0, hence f = ˜ f π B =
0. We have proved that C is an A -small group.(2) Similarly, suppose that ( A n ∣ n < ω ) is an ω -filtration of A . Since A is B -small, there exists n for which Hom ( A / A n , B ) =
0, and so Hom ( A / A i , B ) = i ≥ n . If f ∈ Hom ( A / A i , C ) is nonzero, then π B f ∈ Hom ( A / A i , C / B ) isnonzero because f ( A / A i ) ⊈ B for each i ≥ n . Since there exists k ≥ n for whichHom ( A / A k , C / B ) = ( A / A k , C ) =
0, hence A is C -small. (cid:3) Example 2.4.
The implication of the previous claim cannot be reversed:(1) ∏ p ∈ P Z p is self-small by [21, Theorem 2.5 and Example 2.7], but ⊕ p ∈ P Z p isnot ∏ p ∈ P Z p -small.(2) Since Hom ( Q / Z , Q ) =
0, the group Q / Z is Q -small, but Q / Z is not Q / Z -small. Lemma 2.5.
Let A be an abelian group and M a finite family of abelian groups. (1) If N is A -small for each N ∈ M , then ⊕ M is A -small. (2) If A is N -small for each N ∈ M , then A is ⊕ M -small.Proof. Put M = ⊕ M . Both of the proofs proceed by induction on the cardinalityof M .(1) If ∣M∣ ≤
1, there is nothing to prove. Let the assertion hold true for ∣M∣ − M N = ⊕ M ∖ { N } for arbitrary N ∈ M . Since M N is A -small by theinduction hypothesis, N is A -small by the hypothesis and M / N ≅ M N , we get that M is A -small by Proposition 2.3(1).(2) The same induction argument as in (1) shows A is M -small by Lemma 2.3(2),since A is N -small by the hypothesis and it is M N -small for each N ∈ M by theinduction hypothesis. (cid:3) As the main result of the section we describe which finite sums of relatively smallabelian groups are again relatively small.
Proposition 2.6.
Let M and N be finite families of abelian groups. The followingconditions are equivalent: (1) ⊕ M is ⊕ N -small, (2) M is ⊕ N -small for each M ∈ M , (3) ⊕ M is N -small for each N ∈ N , (4) M is N -small for each M ∈ M and N ∈ N , JOSEF DVOˇR´AK AND JAN ˇZEMLIˇCKA (5) for each M ∈ M , N ∈ N , and ω -filtration ( M i ∣ i < ω ) of M , there exist i < ω with Hom ( M / M i , N ) = .Proof. (1) ⇒ (2) Put F M ∶ = ⊕(M ∖ { M }) ≤ ⊕ M and once (⊕ M)/ F M ≅ M , theclaim follows from Lemma 2.1(1).(1) ⇒ (3) Since N ≤ ⊕ N the assertion is clear by Lemma 2.1(2).(2) ⇒ (4), (3) ⇒ (4) It follows from Lemma 2.1 again.The implication (4) ⇒ (3) is a consequence of Lemma 2.5(1), while the implication(3) ⇒ (1) is shown in Lemma 2.5(2).(4) ⇔ (5) It is an immediate consequence of Proposition 1.7. (cid:3) As a consequence we reformulate [12, Proposition 5]:
Corollary 2.7.
The following conditions are equivalent for a finite family of abeliangroups M and M = ⊕ M : (1) M is self-small, (2) N is N -small for each N , N ∈ M , (3) for every N , N ∈ M and ω -filtration ( M i ∣ i < ω ) of N there exist i < ω with Hom ( N / M i , N ) = . Example 2.8.
Since Hom ( Q , Z ) = Q is self-small and Z is small so Z -smalland Q -small, the group Z ⊕ Q is self-small by Corollary 2.7.3. Self-small products
We start the section by a criterion of self-smallness of a general product (cf. [13,Theorem 5.4]).
Theorem 3.1.
Let M be a family of abelian groups and put M = ∏ M and S = ⊕ M . Then the following conditions are equivalent: (1) M is self-small, (2) M is S -small, (3) M is ⊕ C -small for each countable family C ⊆ M .Proof. The implications (1) ⇒ (2) ⇒ (3) follow from Lemma 2.1(2), since S is embed-dable into M and ⊕ C is embeddable into S .(3) ⇒ (1) Proving indirectly, assume that M is not self-small. Then there existsan ω -filtration ( M i ∣ i < ω ) of M for which Hom ( M / M n , M ) ≠ n < ω byProposition 1.7. Using the same argument as in the proof of Proposition 2.3(2),for each n < ω there exists A n ∈ M such that Hom ( M / M n , A n ) ≠
0. If we put C = { A i ∣ i < ω } , then all A i ’s are embeddable into ⊕ C , hence Hom ( M / M n , ⊕ C) ≠ n < ω , which implies that M is not ⊕ C -small by Proposition 1.7. (cid:3) As A κ is A ( κ ) -small if and only if it is A -small by Corollary 1.5 we obtain thefollowing consequence of Theorem 3.1. Corollary 3.2.
Let A be an abelian group and I a set. Then A I is self-small ifand only if it is A -small. Example 3.3. (1) Q ω is not self-small, since it is an infinitely generated Q -vectorspace, hence it is not Q -small.(2) We have recalled in Example 2.4 that ∏ p ∈ P Z p is self-small, so it is ⊕ p ∈ P Z p -small group by Theorem 3.1. ELF-SMALL PRODUCTS OF ABELIAN GROUPS 7
Let us denote by T A = ⊕ p ∈ P A ( p ) the torsion part of an abelian group A where A ( p ) denotes the p -component of the torsion part. Lemma 3.4.
Let p ∈ P , P be a nonzero p -group, R a nonzero torsion group, T afamily of finite torsion groups, and κ be a cardinal. Then: (1) Z κp is P -small if and only if κ is finite, (2) Z κ is R -small if and only if κ is finite, (3) if ∏ T is P -small, then { T ∈ T ∣ T ( p ) ≠ } is finite, (4) if ∏ T is R -small, then { T ∈ T ∣ T ( p ) ≠ } is finite for each p ∈ P satisfying R ( p ) ≠ .Proof. (1) If κ is finite, then Z κp is finite, and so P -small (it is, in fact, small). If κ isinfinite, then Z κp is an infinitely generated vector space over Z p . Hence infinite directsum of groups Z p , which is not Z p -small, so it is not P -small by Lemma 2.1(2),since there exists Q ≤ P with Q ≃ Z p .(2) It is enough to prove the direct implication. Suppose that κ is infinite. Sincethere exists p ∈ P such that R ( p ) ≠ Z κ /( p Z κ ) ≅ Z κp is not R ( p ) -small by (1).Then Z κ is not R -small by Lemma 2.1(1).(3) Put T p = { T ( p ) ∣ T ∈ T , T ( p ) ≠ } and S = { pS ∣ S ∈ T p } and suppose that κ = ∣ T p ∣ = ∣{ T ∈ T ∣ T ( p ) ≠ }∣ is infinite. Then (∏ T p )/ ∏ S ≅ Z κp which is not P -smallby (1), and so ∏ T p is not P -small by Lemma 2.1(1). Now ∏ T is not P -small byLemma 2.1(1) again, as ∏ T p is a direct summand of ∏ T .(4) It follows from (3) and Lemma 2.1(2). (cid:3) Lemma 3.5.
Let A p be a finite p -group for each p ∈ P . Then ∏ p ∈ P A p is self-small.Proof. Repeating the argument of [21, Lemma 1.7] (cf. also Example 3.3(2)) wecan see that A = ∏ p ∈ P A p / ⊕ p ∈ P A p is divisible, since A q is p -divisible for every p ∈ P except for p = q . Hence A is p -divisible for all primes p and so it is divisible. If f ∈ Hom (∏ p ≠ q A p , A q ) where q ∈ P , then ⊕ p ≠ q A p ⊆ ker f , hence im f is isomorphicto some factor of the divisible group A . Therefore im f is divisible, so im f =
0. Inconsequence, Hom (∏ p ≠ q A p , A q ) = A q is self-small for each q ∈ P implies that ∏ p ∈ P A p is self-small by applying [21, Proposition 1.6]. (cid:3) Now we are ready to describe self-small products of finitely generated groups.
Theorem 3.6.
Let M be a family of nonzero finitely generated abelian groups suchthat at least one N ∈ M has nonzero torsion part and put M = ∏ M , S = ⊕ M and F = S / T S . Then the following conditions are equivalent: (1) M is self-small, (2) S is Z -small and S ( p ) -small for all p ∈ P , (3) S ( p ) is finite for each p ∈ P and S / T S is finitely generated (4) there are only finitely many A ∈ M which are infinite and for each p ∈ P there are only finitely many A ∈ M with A ( p ) ≠ , (5) the family { B ∈ M ∣ Hom ( B, A ) ≠ } is finite for each A ∈ M , (6) there are only finitely many A ∈ M which are infinite and the family { B ∈M ∣ Hom ( C, B ) ≠ } is finite for each finite C ∈ M , (7) M ≅ F ⊕ ∏ p ∈ P M p for a finitely generated free group F and finite abelian p -groups M p for each p ∈ P .Proof. It is well-known that for every finitely generated abelian group A ∈ M thereexists a finitely generated free group F A and a finite torsion group T A such that JOSEF DVOˇR´AK AND JAN ˇZEMLIˇCKA A ≅ F A ⊕ T A . Put F = ⊕ A ∈M F A and T = ⊕ A ∈M T A and note that S ≅ F ⊕ T where F is a free abelian group and T is the torsion part of S . Furthermore M ≅ ∏ A ∈M F A ⊕ ∏ A ∈M T A .(2) ⇔ (3) ⇔ (4) Note that Hom ( T, Z ) =
0, hence S is Z -small if and only if F is Z -small. Thus S is Z -small if and only if S / T S ≅ F is finitely generated which holdstrue if and only if there are only finitely many A ∈ M with nonzero F A , i.e. whichare infinite. Furthermore, it is easy to see that S is S ( p ) -small if and only if S ( p ) isfinite if and only if there exists only finitely many A ∈ M such that A ( p ) ≠ ⇒ (5) Take A ∈ M . Then Hom ( B, A ) ≠ F B ≠ p ∈ P satisfying ( T A ) ( p ) ≠ ≠ ( T B ) ( p ) . Since A is finitely generated, there exist p i ∈ P , i = , . . . , k such that T A = ⊕ ki = ( T A ) ( p i ) . In total, we get { B ∈ M ∣ Hom ( B, A ) ≠ } ⊆ { B ∣ F B ≠ } ∪ k ⋃ i = { B ∣ ( T B ) ( p i ) ≠ } , where both sets on the right-hand side are finite.(5) ⇒ (4) Let A ∈ M be infinite, i.e F A ≠
0. If B is infinite, then Hom ( B, A ) ≠ B ∈ M . Similarly, if A, B ∈ M are such that A ( p ) ≠ ≠ B ( p ) , then Hom ( B, A ) ≠
0, so for each p ∈ P there are onlyfinitely many B ∈ M such that B ( p ) ≠ ⇒ (4) Since M is self-small, it is S -small by Theorem 3.1. Furthermore, ∏ A ∈M F A being a direct summand, hence a factor of M , it is M -small and in conse-quence T -small by Lemma 2.1(2), so ∏ A ∈M F A is finitely generated by Lemma 3.4(2).Therefore there exist only finitely many A with F A ≠
0. Similarly, since ∏ A ∈M T A is T -small, there exist only finitely many A ∈ M such that A ( p ) = ( T A ) ( p ) ≠ p ∈ P by Lemma 3.4(4).(3) ⇒ (7) Note that by (3) F = ⊕ A ∈M F A is finitely generated. Moreover, ∏ A ∈M T A = ∏ A ∈M ⊕ p ∈ P ( T A ) ( p ) ≅ ∏ A ∈M ∏ p ∈ P ( T A ) ( p ) ≅ ∏ p ∈ P ∏ A ∈M ( T A ) ( p ) ≅ ∏ p ∈ P ⊕ A ∈M ( T A ) ( p ) , because T A is finite for all A ∈ M and for each p ∈ P there exist only finitely many A with ( T A ) ( p ) ≠
0. Then M p = ⊕ A ∈M ( T A ) ( p ) is a finite p -group for all p ∈ P and M ≅ F ⊕ ∏ A ∈M T A ≅ F ⊕ ∏ p ∈ P M p (7) ⇒ (1) By Theorem 3.1 it is enough to prove that M is F ⊕ ⊕ p ∈ P M p -small.Since F is finitely generated, it is F ⊕ ⊕ p ∈ P M p -small. As Hom (∏ p ∈ P M p , F ) = ∏ p ∈ P M p is ⊕ p ∈ P M p -small by Proposition 2.6, which holdstrue by Lemma 3.5 and Theorem 3.1.(5) ⇔ (6) The assertion concerning infinite groups follows from the equivalence of(4) and (5). The rest is a consequence of the fact that Hom ( C, B ) ≠ ( B, C ) ≠ B, C . (cid:3) An uncountable cardinal κ is measurable if it admits a κ -additive measure µ ∶ κ → {
0; 1 } such that µ ( κ ) = µ ( x ) = x ∈ κ . A group G is called slender , iffor any homomorphism f ∶ Z ω → G , f ( e i ) = i ∈ ω , where e i denotesthe element of Z ω with π j ( e i ) = δ i,j . Recall that Z is slender by [15, Theorem94.2] and that for a nonmeasurable cardinal κ we have Hom ( Z κ , Z ) ≅ Z ( κ ) by [15,Corollary 94.5] (cf. also [2, Theorem 3.6]). Lemma 3.7. Z κ is Z -small for each cardinal κ .Proof. For finite κ there is nothing to prove, so let us suppose that κ is infi-nite and assume that Z κ is not Z -small. Then there exists a homomorphism ELF-SMALL PRODUCTS OF ABELIAN GROUPS 9 g ∈ Hom ( Z κ , Z ( ω ) ) such that im g is infinitely generated by Proposition 1.7, henceim g is a free abelian group of infinite rank. Since im g ≅ Z ( ω ) is projective, Z ( ω ) isa direct summand of Z κ , i.e. there exists a group A for which Z κ ≅ Z ( ω ) ⊕ A .First, assume that κ = ω . Then Hom ( Z ω , Z ) ≅ Z ( ω ) by [15, Corollary 94.5] as Z is slender by [15, Theorem 94.2]. Hence Z ( ω ) ≅ Hom ( Z ω , Z ) ≅ Hom ( Z ( ω ) ⊕ A, Z ) ≅ Hom ( Z ( ω ) , Z )⊕ Hom ( A, Z ) ≅ Z ω ⊕ Hom ( A, Z ) which is impossible for cardinality reasons (i.e. ∣ Z ( ω ) ∣ < ∣ Z ω ∣ ).We have proved that Z ω is Z -small, so κ > ω . Let λ ≥ κ be a nonmeasurablecardinal (it exists, as for instance each singular cardinal is nonmesurable). ThenHom ( Z λ , Z ) ≅ Z ( λ ) by [15, Corollary 94.5] and Z λ ≅ Z λ ⊕ Z κ as λ + κ = λ , hence Z λ ≅ Z ( ω ) ⊕ B for B = Z λ ⊕ A . We get Z ( λ ) ≅ Hom ( Z λ , Z ) ≅ Hom ( Z ( ω ) ⊕ B, Z ) ≅ Z ω ⊕ Hom ( A, Z ) , which implies that Z ω is embeddable into Z ( λ ) , so it is an infinitely generated freegroup. This contradicts the fact that Z ω is Z -small. (cid:3) Example 3.8.
Expressing Proposition 2.3(1) via the language of short exact se-quences, we can say that relative smallness is transferred from the outer mem-bers to the middle one. The other direction, however, is more complicated: whileLemma 2.1(1) implies the transfer from the middle member to the right, the previ-ous example shows that the transfer to the left does not occur generally: we have Z ( ω ) ↪ Z ω , but Z ( ω ) is not Z -small.Using Corollary 3.2 we can formulate an important consequence: Corollary 3.9. Z κ is self-small for each cardinal κ . We finish the paper by a general criterion of self-small products of finitely gen-erated groups.
Theorem 3.10.
Let M be a family of nonzero finitely generated abelian groupsand put M = ∏ M , S = ⊕ M and F = S / T S . Then the following conditions areequivalent: (1) M is self-small, (2) either T S = , or S ( p ) is finite for each p ∈ P and S / T S is finitely generated (3) either all A ∈ M are free, or the family { B ∈ M ∣ Hom ( B, A ) ≠ } is finitefor each A ∈ M , (4) either M ≅ Z κ for some cardinal κ , or M ≅ F ⊕ ∏ p ∈ P M p for a finitelygenerated free group F and finite abelian p -groups M p for each p ∈ P .Proof. The torsion part of M is zero if and only all groups A ∈ M are free whichmeans that M ≅ Z κ for some cardinal κ is self-small by Corollary 3.9. The casewhen the torsion part of M is nonzero follows directly from Theorem 3.6. (cid:3) Example 3.11.
The finiteness condition in the previous theorem cannot be omittedwithout additional conditions, since e.g. the group Q × ∏ p ∈ P Z p is not self-small by[12, Example 3]. References [1] Albrecht U., Breaz, S.: A note on self-small modules over RM-domains,
J. Algebra Appl.
Bull. Aust. Math. Soc. (2009), No. 2, 205–216.[4] Albrecht, U., Breaz, S., Wickless, W., Purity and self-small groups. Commun. Algebra (2007), No. 11, 3789-3807.[5] Arnold, D.M., Murley, C.E., Abelian groups, A , such that Hom ( A, −) preserves direct sumsof copies of A . Pacific Journal of Mathematics , Vol. (1975), No.1, 7–20.[6] Bass, H.: Algebraic K-theory , Mathematics Lecture Note Series, New York-Amsterdam: W.A.Benjamin, 1968.[7] Breaz S., Self-small abelian groups as modules over their endomorphism rings,
Comm. Algebra
31 (2003), no. 10, 4911–4924.[8] Breaz S.,
A mixed version for a Fuchs’ Lemma . Rend. Sem. Mat. Univ. Padova 144 (2020),61-71.[9] Breaz, S., Schultz, P., Dualities for Self-small Groups
Proc. A.M.S. , (2012), No. 1, 69–82.[10] Breaz, S., ˇZemliˇcka, J. When every self-small module is finitely generated. J. Algebra (2007), 885–893.[11] Colpi, R., Menini, C., On the structure of ⋆ -modules, J. Algebra (1993), 400–419.[12] Dvoˇr´ak, J.: On products of self-small abelian groups , Stud. Univ. Babe¸s-Bolyai Math.
Autocompact objects of Ab5 categories , submitted, 2021,arXiv:2102.04818.[14] Fuchs, L., Infinite Abelian Groups, Vol.I, Academic press, New York and London 1970[15] Fuchs, L.: Infinite Abelian groups. Volume II. New York: Academic Press, 1973.[16] G´omez Pardo, J. L. , Militaru, G., N˘ast˘asescu, C., When is HOM ( M, −) equal to Hom ( M, −) in the category R − gr ?, Comm. Algebra, (1994), 3171–3181.[17] K´alnai, P., Zemliˇcka, J., Compactness in abelian categories , J. Algebra, 534 (2019), 273–288[18] Modoi, C.G.,
Constructing large self-small modules , Stud. Univ. Babe¸s-Bolyai Math.
Abelian categories with applications to rings and modules , 1973, Boston, Aca-demic Press.[20] Rentschler, R.:
Sur les modules M tels que Hom ( M, −) commute avec les sommes directes ,C.R. Acad. Sci. Paris, (1969), 930–933.[21] ˇZemliˇcka, J., When products of self-small modules are self-small. Commun. Algebra (2008), No. 7, 2570–2576. CTU in Prague, FEE, Department of mathematics, Technick´a 2, 166 27 Prague 6 &MFF UK, Department of Algebra, Sokolovsk´a 83, 186 75 Praha 8, Czech Republic
Email address : [email protected] Department of Algebra, Charles University, Faculty of Mathematics and PhysicsSokolovsk´a 83, 186 75 Praha 8, Czech Republic
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