(i) For any ring R, prove that a left R-module B is...

(i) For any ring $R$, prove that a left $R$-module $B$ is injective if and only if $\operatorname{Ext}_{R}^{1}(R / I, B)=\{0\}$ for every left ideal $I$. Hint. Use the Baer criterion. (ii) If $D$ is an abelian group and $\operatorname{Ext}_{\mathbb{Z}}^{1}(\mathbb{Q} / \mathbb{Z}, D)=\{0\}$, prove that $D$ is divisible. The converse is true because divisible abelian groups are injective. Does this hold if we replace $\mathbb{Z}$ by a domain $R$ and $\mathbb{Q} / \mathbb{Z}$ by $\operatorname{Frac}(R) / R ?$

(i) For any ring $R$, prove that a left $R$-module $B$ is injective if and only if $\operatorname{Ext}_{R}^{1}(R / I, B)=\{0\}$ for every left ideal $I$.
Hint. Use the Baer criterion.
(ii) If $D$ is an abelian group and $\operatorname{Ext}_{\mathbb{Z}}^{1}(\mathbb{Q} / \mathbb{Z}, D)=\{0\}$, prove that $D$ is divisible. The converse is true because divisible abelian groups are injective. Does this hold if we replace $\mathbb{Z}$ by a domain $R$ and $\mathbb{Q} / \mathbb{Z}$ by $\operatorname{Frac}(R) / R ?$

Key Concepts

Injective Module

An injective module is a module with the property that every homomorphism from a submodule of any module into it can be extended to a homomorphism from the whole module. This extension property is central in homological algebra and is often used to understand and compute derived functors such as Ext, as well as in decompositions and resolutions of modules.

Baer Criterion

The Baer criterion provides a practical characterization of injective modules. It states that a module is injective if and only if every module homomorphism defined on any left ideal of the ring can be extended to a homomorphism on the entire ring. This criterion is especially useful for proving the injectivity of modules through the vanishing of certain Ext groups.

Ext Functor

The Ext functor measures the extent to which a given module fails to be projective (or injective) and classifies the equivalence classes of module extensions. In particular, the group Ext¹ is used to classify short exact sequences and plays a key role in connecting homological properties of modules with their extension groups. Vanishing of Ext¹ with particular modules, such as quotients by ideals, can characterize injectivity.

Left Ideal and Quotient Modules

Left ideals are specific submodules of a ring that are important in constructing quotient modules. In the context of module theory, quotient modules R/I created from a left ideal I are used in testing the extension properties of a module. They serve as a 'test object' in the application of the Baer criterion to characterize injective modules.

Divisible Module

A divisible module is one in which every element can be divided by any nonzero element of the ring, analogous to divisible abelian groups where every group element is divisible by any nonzero integer. Divisibility often coincides with the injectivity of modules in certain settings, particularly over principal ideal domains, and is critical in understanding the structure of modules and their extension groups.

Field of Fractions for a Domain

The field of fractions of a domain is a construction that turns an integral domain into a field by allowing division by any nonzero element. This concept is important in module theory when considering modules over a domain, particularly in understanding how modules behave when localized and how extension groups relate to the inclusion of the domain into its field of fractions.

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