Let R be a semiprime Goldie ring and let V be a right...

Let $R$ be a semiprime Goldie ring and let $V$ be a right $R$-module. Assume that, for every $v \in V$ and regular element $t \in R$, there exists a unique $v^{\prime} \in V$ with $v^{\prime} t=v .$ Prove that $V$ is injective. In particular, deduce that every $\mathbf{Q}_{\mathrm{cl}}(R)$-module is injective as an $R$-module.

Key Concepts

Semiprime Goldie Ring

A semiprime Goldie ring is a ring with no nonzero nilpotent ideals (semiprime) that also satisfies Goldie’s condition, meaning it does not allow infinite direct sums of nonzero right ideals and has finite uniform dimension. This structure ensures that the classical ring of quotients exists and is semisimple artinian, which is vital for understanding module decompositions and extensions over such rings.

Injective Module

An injective module is one in which every homomorphism from a submodule of any module can be extended to a homomorphism from the whole module. This property is central to homological algebra as it guarantees that the module can serve as a 'test module' for lifting properties in exact sequences, often simplifying the study of extensions and resolutions.

Regular Elements

Regular elements in a ring are those that are not zero divisors, meaning that multiplication by such an element is injective. They play a crucial role in forming the classical ring of quotients by being the elements which can be inverted, thereby enabling division-like operations in both the ring and its modules.

Classical Ring of Quotients

The classical ring of quotients is constructed by inverting all regular elements of a semiprime Goldie ring, yielding a larger ring where every regular element is invertible. This ring often has nicer properties, such as being semisimple artinian, and its modules typically inherit better homological properties, like injectivity, which can then be transferred back to the original ring via restriction of scalars.

Extension and Divisibility Property

An important concept in the problem is the property that for every module element and every regular ring element there is a unique element that acts as a preimage under multiplication. This divisibility-like condition implies a form of local 'invertibility' at the module level and is instrumental in establishing the extendability of morphisms, which is the hallmark of an injective module.

Please give Ace some feedback