Semisimple Is Equivalent To Radical Is Trivial

Abstract Algebra: A Fundamental Concept

In the realm of abstract algebra, particularly in the study of modules, two crucial concepts are semisimple and radical. A semisimple module is one that can be expressed as a direct sum of simple modules, while the radical of a module is the intersection of all its maximal submodules. In this article, we will delve into the relationship between these two concepts and prove that a module satisfying the descending chain condition (DCC) is semisimple if and only if its radical is trivial.

Descending Chain Condition: A Crucial Assumption

Before we begin, let's recall the definition of the descending chain condition. A module $M$ over a ring $R$ satisfies the DCC if for any sequence of submodules $M_1 \supseteq M_2 \supseteq M_3 \supseteq \ldots$, there exists a positive integer $n$ such that $M_n = M_{n+1} = M_{n+2} = \ldots$. In other words, the chain of submodules eventually stabilizes.

Semisimple Modules: A Direct Sum of Simple Modules

A module $M$ is said to be semisimple if it can be expressed as a direct sum of simple modules. A simple module is a module that has no proper non-trivial submodules. In other words, the only submodules of a simple module are the zero submodule and the module itself.

Radical of a Module: The Intersection of Maximal Submodules

The radical of a module $M$, denoted by $\operatorname{Rad}(M)$, is the intersection of all maximal submodules of $M$. A maximal submodule is a proper submodule that is not contained in any other proper submodule.

Trivial Radical: A Module with No Non-Trivial Maximal Submodules

A module $M$ has a trivial radical if $\operatorname{Rad}(M) = 0$. In other words, the only maximal submodule of $M$ is the zero submodule itself.

The Main Result: Semisimple is Equivalent to Radical is Trivial

We are now ready to state the main result:

Theorem: Let $M$ be a module over a ring $R$ that satisfies the descending chain condition. Then, $M$ is semisimple if and only if its radical is trivial.

Proof:

Necessity

Suppose $M$ is semisimple. Then, $M$ can be expressed as a direct sum of simple modules:

$$M = M_1 \oplus M_2 \oplus \ldots \oplus M_n$$

where each $M_i$ is a simple module. Since each $M_i$ is simple, it has no proper non-trivial submodules. Therefore, the only maximal submodule of $M$ is the zero submodule itself, which implies that $\operatorname{Rad}(M) = 0$. Hence, the radical of $M$ is trivial.

Sufficiency

Conversely, suppose that the radical of $M$ is trivial, i.e., $\operatorname{Rad}(M) = 0$. We need to show that $M$ is semisimple. Let $N$ be a submodule of $M$. We want to show that $N$ is a direct summand of $M$. Since $M$ satisfies the DCC, the chain of submodules $N \supseteq \operatorname{Rad}(N) \supseteq \operatorname{Rad}^2(N) \supseteq \ldots$ eventually stabilizes. Let $k$ be the smallest positive integer such that $\operatorname{Rad}^k(N) = \operatorname{Rad}^{k+1}(N)$. Then, $\operatorname{Rad}^k(N)$ is a maximal submodule of $N$. Since $\operatorname{Rad}(M) = 0$, we have $\operatorname{Rad}^k(N) = 0$. Therefore, $N$ is a simple module. Since $M$ is a direct sum of simple modules, we conclude that $M$ is semisimple.

Conclusion

In this article, we have proved that a module satisfying the descending chain condition is semisimple if and only if its radical is trivial. This result highlights the importance of the radical in the study of modules and provides a useful tool for analyzing the structure of modules.

References

• [1] Hungerford, T. W. (1974). Algebra. Springer-Verlag.
• [2] Lang, S. (1993). _Algebra**. Springer-Verlag.
• [3] Rotman, J. J. (1995). _An Introduction to Homological Algebra**. Springer-Verlag.

Further Reading

• [1] Bourbaki, N. (1958). _Algebre**. Hermann.
• [2] Cartan, H., and Eilenberg, S. (1956). _Homological Algebra**. Princeton University Press.
• [3] MacLane, S. (1963). _Homology**. Springer-Verlag.
  Q&A: Semisimple is Equivalent to Radical is Trivial =====================================================

Frequently Asked Questions

In this article, we will address some of the most common questions related to the concept of semisimple modules and their relationship with the radical.

Q: What is the difference between a semisimple module and a simple module?

A: A simple module is a module that has no proper non-trivial submodules. On the other hand, a semisimple module is a module that can be expressed as a direct sum of simple modules.

Q: Why is the radical of a module important?

A: The radical of a module is the intersection of all its maximal submodules. It is an important concept in the study of modules because it provides information about the structure of the module.

Q: What is the significance of the descending chain condition in this context?

A: The descending chain condition is a crucial assumption in this article. It ensures that the chain of submodules eventually stabilizes, which is necessary for the proof of the main result.

Q: Can you provide an example of a module that satisfies the descending chain condition but is not semisimple?

A: Yes, consider the module $\mathbb{Z}/4\mathbb{Z}$ over the ring $\mathbb{Z}$. This module satisfies the descending chain condition, but it is not semisimple because it has a non-trivial radical.

Q: How does the radical of a module relate to the socle of a module?

A: The socle of a module is the sum of all its simple submodules. The radical of a module is the intersection of all its maximal submodules. While the socle and radical are related concepts, they are not the same.

Q: Can you provide an example of a module that has a non-trivial radical but is not semisimple?

A: Yes, consider the module $\mathbb{Z}/6\mathbb{Z}$ over the ring $\mathbb{Z}$. This module has a non-trivial radical, but it is not semisimple because it has a non-trivial maximal submodule.

Q: How does the main result of this article relate to other results in module theory?

A: The main result of this article is a fundamental result in module theory that highlights the importance of the radical in the study of modules. It has implications for other results in module theory, such as the study of injective and projective modules.

Q: Can you provide a proof of the main result using a different approach?

A: Yes, there are several different approaches to proving the main result. One alternative approach is to use the concept of injective modules and the fact that a module is semisimple if and only if it is injective.

Conclusion

In this article, we have addressed some of the most common questions related to the concept of semisimple modules and their relationship with the radical. We hope that this article has provided a useful resource for students and researchers in the field of abstract algebra.

References

• [1] Hungerford, T. W. (1974). Algebra. Springer-Verlag.
• [2] Lang, S. (1993). _Algebra**. Springer-Verlag.
• [3] Rotman, J. J. (1995). _An Introduction to Homological Algebra**. Springer-Verlag.

Further Reading

• [1] Bourbaki, N. (1958). _Algebre**. Hermann.
• [2] Cartan, H., and Eilenberg, S. (1956). _Homological Algebra**. Princeton University Press.
• [3] MacLane, S. (1963). _Homology**. Springer-Verlag.