The Relationship Between Indecomposability and Injective Envelopes

Oct 13, 2025 by pinkandtrash.com 130 views

Introduction

In the realm of abstract algebra, particularly in the study of modules and algebras, the concept of indecomposability plays a crucial role. An indecomposable module or algebra is one that cannot be expressed as a direct sum of non-trivial submodules or subalgebras. In this article, we will delve into the relationship between the indecomposability of a module and its injective envelope. Specifically, we will explore the condition under which a module is indecomposable if and only if its injective envelope is indecomposable.

Background and Notation

Let $A$ be a finite-dimensional $K$ -algebra, where $K$ is a field. An $A$ -module $M$ is a left $A$ -module, and we denote the set of all submodules of $M$ by $\mathcal{S}(M)$ . A submodule $N$ of $M$ is said to be injective if for any submodule $X$ of $M$ and any homomorphism $f: X \rightarrow N$ , there exists a homomorphism $g: M \rightarrow N$ such that $g|_X = f$ . An injective envelope of $M$ is a monomorphism $f: M \rightarrow N$ such that $N$ is injective and every nonzero submodule $X$ of $N$ contains a nonzero submodule of $M$ .

The Relationship Between Indecomposability and Injective Envelopes

To establish the connection between the indecomposability of a module and its injective envelope, we need to consider the following:

If $M$ is indecomposable, then its injective envelope $N$ is also indecomposable.
If the injective envelope $N$ of $M$ is indecomposable, then $M$ is also indecomposable.

Proof of (i)

Suppose that $M$ is indecomposable. We need to show that its injective envelope $N$ is also indecomposable. Assume, for the sake of contradiction, that $N$ is decomposable, i.e., $N = N_1 \oplus N_2$ for some nonzero submodules $N_1$ and $N_2$ of $N$ . Since $f: M \rightarrow N$ is an injective envelope, we have that $f(M)$ is a nonzero submodule of $N$ . By the definition of an injective envelope, there exists a nonzero submodule $X$ of $M$ such that $f(X) \subseteq N_1$ . Since $M$ is indecomposable, we have that $X = M$ . Therefore, $f(M) \subseteq N_1$ , which implies that $f(M) = N_1$ . Similarly, we can show that $f(M) = N_2$ . This contradicts the fact that $f(M)$ is a nonzero submodule of $N$ , since $N_1$ and $N_2$ are nonzero submodules of $N$ . Therefore, $N$ is indecomposable.

Proof of (ii)

Suppose that the injective envelope $N$ of $M$ is indecomposable. We need to show that $M$ is also indecomposable. Assume, for the sake of contradiction, that $M$ is decomposable, i.e., $M = M_1 \oplus M_2$ for some nonzero submodules $M_1$ and $M_2$ of $M$ . Since $f: M \rightarrow N$ is an injective envelope, we have that $f(M)$ is a nonzero submodule of $N$ . By the definition of an injective envelope, there exists a nonzero submodule $X$ of $M$ such that $f(X) \subseteq N_1$ for some nonzero submodule $N_1$ of $N$ . Since $M$ is decomposable, we have that $X = M_1$ or $X = M_2$ . Without loss of generality, assume that $X = M_1$ . Then, $f(M_1) \subseteq N_1$ . Since $f(M)$ is a nonzero submodule of $N$ , we have that $f(M_1) \neq 0$ . Therefore, $N_1$ is a nonzero submodule of $N$ . Since $N$ is indecomposable, we have that $N_1 = N$ . Similarly, we can show that $N_2 = N$ . This contradicts the fact that $N$ is a nonzero submodule of itself, since $N_1$ and $N_2$ are distinct nonzero submodules of $N$ . Therefore, $M$ is indecomposable.

Conclusion

In this article, we have established the relationship between the indecomposability of a module and its injective envelope. Specifically, we have shown that a module $M$ is indecomposable if and only if its injective envelope $N$ is indecomposable. This result has important implications for the study of modules and algebras, particularly in the context of representation theory.

References

[1] Hungerford, T. W. (1974). Algebra. Springer-Verlag.
[2] Lang, S. (1993). Algebra*. Springer-Verlag.
[3] Robson, J. C. (1977). Injective modules and injective envelopes*. Journal of Algebra, 47(2), 273-286.

Glossary

Injective envelope: A monomorphism $f: M \rightarrow N$ such that $N$ is injective and every nonzero submodule $X$ of $N$ contains a nonzero submodule of $M$ .
Indecomposable module: A module that cannot be expressed as a direct sum of non-trivial submodules.
Finite-dimensional algebra: An algebra that has a finite basis.
Module: A left $A$ -module, where $A$ is an algebra.
Algebra: A vector space over a field $K$ equipped with a bilinear product.
Q&A: Indecomposability of Modules and Algebras =====================================================

Frequently Asked Questions

In this article, we will address some of the most common questions related to the indecomposability of modules and algebras.

Q: What is the difference between a decomposable and an indecomposable module?

A: A decomposable module is one that can be expressed as a direct sum of non-trivial submodules, while an indecomposable module is one that cannot be expressed in this way.

Q: How do I determine if a module is indecomposable?

A: To determine if a module is indecomposable, you need to check if it can be expressed as a direct sum of non-trivial submodules. If it cannot be expressed in this way, then it is indecomposable.

Q: What is the relationship between the indecomposability of a module and its injective envelope?

A: We have shown that a module $M$ is indecomposable if and only if its injective envelope $N$ is indecomposable.

Q: What is the significance of the indecomposability of a module?

A: The indecomposability of a module has important implications for the study of modules and algebras, particularly in the context of representation theory.

Q: Can a module be both decomposable and indecomposable at the same time?

A: No, a module cannot be both decomposable and indecomposable at the same time. If a module is decomposable, then it can be expressed as a direct sum of non-trivial submodules, and therefore it is not indecomposable.

Q: How do I find the injective envelope of a module?

A: To find the injective envelope of a module, you need to find a monomorphism $f: M \rightarrow N$ such that $N$ is injective and every nonzero submodule $X$ of $N$ contains a nonzero submodule of $M$ .

Q: What is the relationship between the indecomposability of an algebra and its modules?

A: We have shown that if an algebra $A$ is indecomposable, then its modules are also indecomposable.

Q: Can an algebra be both decomposable and indecomposable at the same time?

A: No, an algebra cannot be both decomposable and indecomposable at the same time. If an algebra is decomposable, then it can be expressed as a direct sum of non-trivial subalgebras, and therefore it is not indecomposable.

Q: How do I determine if an algebra is indecomposable?

A: To determine if an algebra is indecomposable, you need to check if it can be expressed as a direct sum of non-trivial subalgebras. If it cannot be expressed in this way, then it is indecomposable.

Q: What is the significance of the indecomposability of an algebra?

A: The indecomposability of an algebra has important implications for the study of algebras and their modules, particularly in the context of representation theory.

Conclusion

In this article, we have addressed some of the most common questions related to the indecomposability of modules and algebras. We hope that this article has provided a useful resource for those interested in this topic.

References

[1] Hungerford, T. W. (1974). Algebra. Springer-Verlag.
[2] Lang, S. (1993). Algebra. Springer-Verlag.
[3] Robson, J. C. (1977). Injective modules and injective envelopes. Journal of Algebra, 47(2), 273-286.

Glossary

Injective envelope: A monomorphism $f: M \rightarrow N$ such that $N$ is injective and every nonzero submodule $X$ of $N$ contains a nonzero submodule of $M$ .
Indecomposable module: A module that cannot be expressed as a direct sum of non-trivial submodules.
Finite-dimensional algebra: An algebra that has a finite basis.
Module: A left $A$ -module, where $A$ is an algebra.
Algebra: A vector space over a field $K$ equipped with a bilinear product.