(i) Prove that any family of R-maps (fj: Uj →Vj)j ∈J can be assembled into an R-map φ: ⊕j Uj →⊕j

step 1 In this problem we're asked to first show that asymmetric difference B is equal to B, symmetric

(i) Prove that any family of R-maps (fj: Uj →Vj)j ∈J can be...

(i) Prove that any family of $R$-maps $\left(f_{j}: U_{j} \rightarrow V_{j}\right)_{j \in J}$ can be assembled into an $R$-map $\varphi: \bigoplus_{j} U_{j} \rightarrow \bigoplus_{j} V_{j}$, namely, $\varphi:\left(u_{j}\right) \mapsto\left(f_{j}\left(u_{j}\right)\right)$. (ii) Prove that $\varphi$ is an injection if and only if each $f_{j}$ is an injection.

(i) Prove that any family of $R$-maps $\left(f_{j}: U_{j} \rightarrow V_{j}\right)_{j \in J}$ can be assembled into an $R$-map $\varphi: \bigoplus_{j} U_{j} \rightarrow \bigoplus_{j} V_{j}$, namely, $\varphi:\left(u_{j}\right) \mapsto\left(f_{j}\left(u_{j}\right)\right)$.
(ii) Prove that $\varphi$ is an injection if and only if each $f_{j}$ is an injection.

Key Concepts

Componentwise Map Construction

Constructing a map by applying given functions to the corresponding components of a direct sum is a standard technique in algebra. In this method, a family of maps, each acting on a component of the direct sum, is assembled into a single map on the entire direct sum by defining its action on each coordinate individually. This approach illustrates how operations defined on simpler parts can be extended to more complex structures.

Injectivity Condition

Injectivity in the context of module homomorphisms means that the only element mapping to the zero element is the zero element itself, which is a key property ensuring that the map is one-to-one. When assembling a map from a family of maps defined on a direct sum, the resulting map is injective if and only if each individual component map is injective. This relationship highlights how global properties of a constructed map depend on the corresponding properties of its constituent parts.

Direct Sum of Modules

The direct sum is a construction in module theory (or, more generally, in abelian categories) that takes a family of modules and forms a new module whose elements are tuples with entries from each module, with almost all entries being zero. This structure enables one to work with multiple modules in a unified manner while preserving their individual properties, and it plays a key role in decomposing and assembling modules.

Module Homomorphism

A module homomorphism (or R-map) is a function between modules that preserves the module operations, namely addition and scalar multiplication. This concept is foundational in understanding the structure of modules, as it facilitates the study of maps between algebraic structures, ensuring the operations remain consistent under the mapping.

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