Let W ⊆V be R-modules. If W and V / W are nonsingular, prove...

Let $W \subseteq V$ be $R$-modules. If $W$ and $V / W$ are nonsingular, prove that $V$ is nonsingular. Show that $(\operatorname{Sing}(V)+W) / W \subseteq \operatorname{Sing}(V / W)$ but that equality need not occur.

Key Concepts

Singular Submodule

In module theory, the singular submodule of a module consists of all elements that are 'annihilated' in a strong sense, meaning that each element is annihilated by an essential right ideal of the ring. This concept captures those elements that exhibit degenerate behavior in terms of their interaction with the ring, and is critical when defining or assessing the overall 'quality' of a module's structure.

Nonsingular Module

A module is called nonsingular if its singular submodule is trivial (i.e., consists only of the zero element). This property indicates that every nonzero element of the module behaves non-trivially under the ring action, ensuring that the module does not contain any pathological or degenerative elements. Proving a module is nonsingular often involves showing that any potential singular elements must vanish.

Quotient Module

A quotient module, formed by taking the quotient of a module V by one of its submodules W, is central to many arguments in module theory. The structure of V/W reflects how the original module V is 'modulated' by the submodule W. The behavior of singular elements under the passage from V to V/W (and vice versa) is a key aspect in proving or disproving nonsingularity properties.

Submodule Sum and Projection

The operation of taking the sum of submodules, particularly when coupled with projection onto a quotient module, is an important tool in module theory. In this context, one studies the image of the sum (Sing(V)+W) under the canonical projection to V/W, revealing how the singular submodule of V relates to the singular submodule of the quotient module V/W. This analysis shows that while the image is always contained within Sing(V/W), equality is not guaranteed, reflecting the subtle interplay between module components.

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