Let R be a ring. Prove directly, without using Theorem 22.4,...

Let $R$ be a ring. Prove directly, without using Theorem $22.4$, that the following are equivalent.
i. All right $R$-modules are injective.
ii. All right ideals of $R$ are injective.
iii. $R$ is a Wedderburn ring.

Key Concepts

Injective Module

An injective module is a module with the property that any homomorphism defined on a submodule can be extended to the entire module. This extension property is central to homological algebra and characterizes modules that are 'flexible' in terms of their embedding into larger modules, often implying strong decomposability properties.

Ring

A ring is an algebraic structure consisting of a set equipped with two binary operations—addition and multiplication—that satisfy certain axioms such as associativity, distributivity, and the existence of an additive identity and additive inverses. Rings provide the basic framework in which module theory and concepts like ideals are defined.

Module

A module over a ring generalizes the idea of a vector space by allowing the scalars to come from a ring instead of a field. This structure permits the study of linear algebra in a broader context and is crucial for exploring extension properties, such as injectivity, of module homomorphisms.

Right Ideal

A right ideal is a subset of a ring that is an additive subgroup and is closed under multiplication from the right by any element of the ring. Right ideals play fundamental roles in understanding the structure of a ring, especially through their relationship to modules and quotient constructions.

Wedderburn Ring

A Wedderburn ring is a ring that is semisimple, meaning it can be decomposed as a finite direct sum of simple modules or, equivalently, is isomorphic to a direct product of matrix rings over division rings. This structure underpins many deep results in ring theory, including the fact that every module (or every right module) over such a ring is both injective and projective.

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