(i) If Zi ≅ℤ for all i, prove that Homℤ(∏i=1^∞ Zi, ℤ) ¥ ∏i=1^∞ Homℤ(Zi, ℤ) Hint. A theorem of J.

step 1 So, if x1, x2 belongs to k, then there are, there are n1, comma, n2 belonging to n, such that x

(i) If Zi ≅ℤ for all i, prove that Homℤ(∏i=1^∞ Zi, ℤ) ¥...

(i) If $Z_{i} \cong \mathbb{Z}$ for all $i$, prove that $$ \operatorname{Hom}_{\mathbb{Z}}\left(\prod_{i=1}^{\infty} Z_{i}, \mathbb{Z}\right) ¥ \prod_{i=1}^{\infty} \operatorname{Hom}_{\mathbb{Z}}\left(Z_{i}, \mathbb{Z}\right) $$ Hint. A theorem of J. Los and, independently, of E. C. Zeeman (see Fuchs, Infinite Abelian Groups II, Section 94) says that (ii) Let $p$ be a prime and let $B_{n}$ be a cyclic group of order $p^{n}$, where $n$ is a positive integer. If $A=\bigoplus_{n=1}^{\infty} B_{n}$, prove that $$ \operatorname{Hom}_{k}\left(A, \bigoplus_{n=1}^{\infty} B_{n}\right) \neq \bigoplus_{n=1}^{\infty} \operatorname{Hom}_{k}\left(A, B_{n}\right) $$ Hint. Prove that $\operatorname{Hom}(A, A)$ has an element of infinite order, while every element in $\bigoplus_{n=1}^{\infty} \operatorname{Hom}_{k}\left(A, B_{n}\right)$ has finite order. (iii) Prove that Hom $_{2}\left(\prod_{n \geq 2} \mathbb{I}_{n}, \mathbb{Q}\right) \neq \prod_{n \geq 2} \operatorname{Hom}_{\mathbb{Z}}\left(\mathbb{I}_{n}, \mathbb{Q}\right)$.

(i) If $Z_{i} \cong \mathbb{Z}$ for all $i$, prove that
$$
\operatorname{Hom}_{\mathbb{Z}}\left(\prod_{i=1}^{\infty} Z_{i}, \mathbb{Z}\right) ¥ \prod_{i=1}^{\infty} \operatorname{Hom}_{\mathbb{Z}}\left(Z_{i}, \mathbb{Z}\right)
$$
Hint. A theorem of J. Los and, independently, of E. C. Zeeman (see Fuchs, Infinite Abelian Groups II, Section 94) says that
(ii) Let $p$ be a prime and let $B_{n}$ be a cyclic group of order $p^{n}$, where $n$ is a positive integer. If $A=\bigoplus_{n=1}^{\infty} B_{n}$, prove that
$$
\operatorname{Hom}_{k}\left(A, \bigoplus_{n=1}^{\infty} B_{n}\right) \neq \bigoplus_{n=1}^{\infty} \operatorname{Hom}_{k}\left(A, B_{n}\right)
$$
Hint. Prove that $\operatorname{Hom}(A, A)$ has an element of infinite order, while every element in $\bigoplus_{n=1}^{\infty} \operatorname{Hom}_{k}\left(A, B_{n}\right)$ has finite order.
(iii) Prove that Hom $_{2}\left(\prod_{n \geq 2} \mathbb{I}_{n}, \mathbb{Q}\right) \neq \prod_{n \geq 2} \operatorname{Hom}_{\mathbb{Z}}\left(\mathbb{I}_{n}, \mathbb{Q}\right)$.

Key Concepts

Direct Sum vs. Direct Product

In module theory and the study of abelian groups, the direct sum and direct product serve as two distinct types of constructions. The direct sum of modules (or groups) restricts to those tuples with only finitely many nonzero entries, while the direct product allows arbitrary (even infinitely many nonzero) entries. This distinction becomes crucial in infinite settings, as certain properties—like behavior under Hom and duality functors—differ markedly between the two constructions.

Hom Functor and Dual Modules

The Hom functor assigns to each module (or abelian group) A the group Hom(A, B) of all module homomorphisms from A to a fixed module B. This construction, when B is the ring or field of scalars, often produces the 'dual module.' Understanding how this functor interacts with direct sums and products—namely that Hom of a direct sum is naturally isomorphic to the direct product of Homs, but not necessarily so for a direct product—is fundamental in homological algebra and the study of infinite abelian groups.

Functoriality with Respect to Direct Sums and Products

In category theory and module theory, the Hom functor is contravariant (or covariant, depending on the context) and interacts differently with direct sums and products. Specifically, while Hom turns direct sums into direct products in many situations, applying the functor to a direct product does not simply decompose into a product of Hom groups. This discrepancy highlights important functorial properties such as limits and colimits and is central to understanding the structure of mappings in infinite-dimensional contexts.

Torsion versus Torsion-Free Elements

The concept distinguishes modules or abelian groups in which every element has finite order (torsion groups) from those where at least one nonzero element has infinite order (torsion?free). This distinction is crucial when studying Hom groups because the existence of homomorphisms with infinite order can signify that the dual of an infinite abelian group possesses structural properties different from those of a direct sum of simpler components, thereby affecting the Hom functor's outcome.

Order of Elements in Hom Groups

When dealing with Hom groups, the order of its elements—whether finite or infinite—plays an important role in their structure and classification. In many cases, as demonstrated in specific examples, an element in a Hom group may have infinite order while every element in a related direct sum of Hom groups might have finite order. Recognizing and proving such differences is vital in understanding why certain homomorphism groups may not decompose as one might initially expect.

Infinite Abelian Group Structure

Infinite abelian groups require careful analysis since many properties evident in finitely generated groups fail in the infinite setting. The interplay between various constructions—such as direct sums, direct products, and dual groups—along with the examination of element orders and torsion properties, forms the backbone of the study of infinite abelian groups. This structural insight is key to discerning and proving relationships between different Hom groups.

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