On the Extensions of Zassenhaus Lemma and Goursat’s Lemma to Algebraic Structures

Meng, Fanning; Guo, Junhui

doi:https://doi.org/10.1155/2022/7705500

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Volume 2022 | Article ID 7705500 | https://doi.org/10.1155/2022/7705500

On the Extensions of Zassenhaus Lemma and Goursat’s Lemma to Algebraic Structures

Fanning Meng¹and Junhui Guo¹

Academic Editor: Li Guo

Received15 Oct 2021

Revised26 Nov 2021

Accepted28 Dec 2021

Published03 Feb 2022

Abstract

The Jordan–Hölder theorem is proved by using Zassenhaus lemma which is a generalization of the Second Isomorphism Theorem for groups. Goursat’s lemma is a generalization of Zassenhaus lemma, it is an algebraic theorem for characterizing subgroups of the direct product of two groups , and it involves isomorphisms between quotient groups of subgroups of and . In this paper, we first extend Goursat’s lemma to -algebras, i.e., give the version of Goursat’s lemma for algebras, and then generalize Zassenhaus lemma to rings, -modules, and -algebras by using the corresponding Goursat’s lemma, i.e., give the versions of Zassenhaus lemma for rings, -modules, and -algebras, respectively.

1. Introduction

The Fundamental Homomorphism Theorem (or the First Isomorphism Theorem) provided by Noether [1] in 1927 shows that every homomorphism gives rise to an isomorphism and that quotient groups are merely constructions of homomorphic images. Even it has simple form, it expresses the important properties of quotient group. Noether emphasized the fundamental importance of this fact, and it has been widely used in the field of universal algebra and to prove the existence of some natural isomorphisms. The Diamond Isomorphism Theorem (or the Second Isomorphism Theorem) which is the consequence of the Fundamental Homomorphism Theorem is formulated in terms of subgroups of the normalizer and relates two quotient groups involving products and intersections of subgroups. After that, many researchers generalized the Second Isomorphism Theorem to other structures, such as Tamaschke [2, 3] generalized it to Schur semigroups [4] and Endam and Vilela [5] extended it to -algebras introduced by Neggers and Kim [6, 7]. Sequentially, the First Isomorphism Theorem and Second Isomorphism Theorem are generalized to rings, vector spaces, -modules, and -algebras, respectively. In this paper, we suppose that the readers are familiar with the structures of groups, rings, -modules, and -algebras.

In 1934, Zassenhaus [8] found a new and beautiful proof of the Jordan–Hölder theorem via Zassenhaus lemma. Zassenhaus lemma (also called Butterfly lemma) is well known in group theory as a generalization of the Second Isomorphism Theorem for groups. Subsequently, many researchers extended Zassenhaus lemma to other structures, such as Teh [9, 10] extended it to universal algebras via employment of the graph theory, Wyler [11] extended it to categories under conditions, Ngaha Ngaha [12] considered the Second Isomorphism Theorem and Zassenhaus lemma in star-regular categories and described Zassenhaus lemma in the category of commutative Hopf algebras, and Prathomjit et al. [13] extended Zassenhaus lemma and the Schreier refinement theorem to the case of gyrogroups and then used these results to prove the Jordan–Hölder theorem for gyrogroups.

On the other hand, as we know, there is a natural question in group theory, that is, how many subgroups do exist for the general group ? If we want to count the number of subgroups of , we should first express the form of subgroups of . In 1889, Goursat [14] gave Goursat’s lemma which is an algebraic theorem for characterizing subgroups of the direct product of two groups , and involves isomorphisms between quotient groups of subgroups of and . After that, many researchers expressed the subgroups in certain classes of groups; for instance, Usenko [15] introduced a reduced crossed homomorphism and used it to describe the subgroups of a semidirect product of groups and characterized the subsemidirect products and semidirect products with a given structure of normal subgroups. Bauer et al. [16] obtained a description of the subgroups of a direct product of a finite number of groups by proving a generalization of Goursat’s lemma. In 2009, Anderson and Camillo [17] proved Zassenhaus lemma by using Goursat’s lemma, which means that Zassenhaus lemma is a corollary of Goursat’s lemma. From Goursat’s lemma, one can recover a more general version of Zassenhaus lemma. According to the above, in this paper, our main aim is to extend Goursat’s lemma to -algebras and give the expression of the form of subalgebras of general -algebras. Further, we generalize Zassenhaus lemma to rings, -modules, and -algebras by using the corresponding Goursat’s lemma, respectively, and show that the corresponding results are the generalization of the Second Isomorphism Theorem for rings, -modules, and -algebras, respectively.

In fact, Goursat’s lemma (see Theorem 1) gave a way to describe the subgroups of a direct product which involves isomorphisms between quotient groups of subgroups of and , i.e.,is a subgroup of , where are subgroups of such that for , and is a group isomorphism. Subsequently, Anderson and Camillo (see Theorem 2) described the subrings of a direct product which involves isomorphisms between quotient rings of subrings of and , i.e.,is a subring of , where are subrings of such that is an ideal of for , and is a ring isomorphism. For -modules, Dickson (see Theorem 3) described the submodules of a direct product which involves isomorphisms between quotient -modules of submodules of and , i.e.,is a submodule of , where are -submodules of such that for , and is a -module isomorphism.

Since an -algebra has the ring structure and -module structure, as a generalization, we consider Goursat’s lemma for -algebras (see Corollary 1) and describe the subalgebras of a direct product which involves isomorphisms between quotient -algebras of subalgebras of and , that is,is a subalgebra of , where are subalgebras of such that is algebraic ideal of for , and is an -algebra isomorphism. Indeed, Lambek [18] gave Goursat’s characterization of the subgroups of the direct product of two groups (also for a general class of algebras) under conditions by using graph theory. In this paper, we give a different form for the subalgebras of -algebras. However, for other structures, such as Lie algebras, quantum cluster algebras [19], -algebras, and free differential algebras [20], we still do not know their Goursat’s characterization.

Furthermore, Anderson and Camillo [17] used Goursat’s lemma to prove Zassenhaus lemma for groups (see Theorem 4) which is stated as follows:where are subgroups of such that and . As a generalization, we consider Zassenhaus lemma for rings (see Theorem 5), -modules (see Theorem 6), and -algebras (see Theorem 7), respectively, and obtain the following results:(1)If are subrings of a ring such that is an ideal of for , then(2)If are submodules of -module satisfying for commutative ring with identity, then(3)If is an -algebra and are subalgebras of such that is an algebraic ideal of for , then

This paper is organized as follows. In Section 2, we first complete the proof of Goursat’s lemma for -modules and then extend Goursat’s lemma to -algebras, i.e., give the version of Goursat’s lemma for algebras and give the form of subalgebras of -algebras, which is different from Lambek’s form [18]. Further, since Zassenhaus lemma is used to prove the Jordan–Hölder theorem and it is also the generalization the Second Isomorphism Theorem for groups, as a generalization, we extend Zassenhaus lemma to rings, -modules, vector spaces, and -algebras in terms of algebraic ideal in Section 3, i.e., give the versions of Zassenhaus lemma for rings, -modules, and -algebras, respectively.

2. Goursat’s Lemma for Groups, Rings, -Modules, and -Algebras

Anderson and Camillo [17] gave an exposition of Goursat’s lemma for groups and rings, and Dickson [21] gave Goursat’s lemma for -modules without proof, respectively. In this section, we complete the proof of Goursat’s lemma for -modules and give an exposition of Goursat’s lemma for -algebras as a corollary.

Theorem 1. (Goursat’s lemma for groups, Theorem 4 in [17]). Let be groups.(1)Let be a subgroup of , and Then, are subgroups of with for , and the map is an isomorphism, where . Moreover, if , then and , the center of .(2)Let be subgroups of with for , and let be an isomorphism. Then, is a subgroup of . Furthermore, suppose and for , then .(3)The constructions given in (1) and (2) are inverses to each other.

Theorem 2. (Goursat’s lemma for rings, Theorem 11 in [17]). Let be rings.(1)Let be a subring of , and Then, is a subring of and is an ideal of for . Moreover, the map defined by for is a ring isomorphism.(2)Suppose that is a subring of and is an ideal of for , and is a ring isomorphism. Then, is a subring of .(3)The construction given in (1) and (2) is inverse to each other.In Theorem 4 in [17], the authors stated Goursat’s lemma for -modules without proof (also see [21, 22]). In the following, we provide a proof of Goursat’s lemma for -modules by using the submodule criterion [23].

Theorem 3. (Goursat’s lemma for -modules). Let be a commutative ring with identity and are -modules.(1)Let be a submodule of , and then are submodules of with for , and the map given by is a -module isomorphism, where .(2)Suppose that are submodules of with for and the map is a -module isomorphism, then is a submodule of .(3)The construction given in (1) and (2) is inverse to each other.

Proof. (1) From Theorem 1, it is obvious that for . Since is a submodule of , we havefor any and any . This means that . Thus, is a submodule of following the submodule criterion [23]. On the other hand, for any , there exist such that . Hence,for any . It follows that , and then is a submodule of . Similarly, using the submodule criterion, we can obtain that are submodules of easily.
For the map given by , where , it is clear that are -modules and is an additive group isomorphism by Theorem 1. So, it is enough to prove that for any and . Since for any , and , we have . Hence, is a -module isomorphism.
(2) Suppose that are submodules of with for , and the map is a -module isomorphism. Since , we have for any , andThis means that . Therefore, is a submodule of by the submodule criterion.
(3) From the proofs of (1) and (2), we can easily obtain (3).
It is well known that an -algebra has the ring structure and -module structure concurrently, and the operations of these two structures are compatible, i.e., for any and . We firstly introduce some definitions about -algebras for the commutative ring with identity.

Definition 1. (see [23–25]). Let be a commutative ring with identity. An -algebra is a ring with identity together with a ring homomorphism mapping to such that the subring of is contained in the center of , i.e., commutes with every element of for each . A subalgebra of an -algebra is a subring of and a submodule of . A left (respectively, right, two sided) algebraic ideal of an -algebra is a left (respectively, right, two sided) ideal of the ring and a submodule of .
If are two -algebras, an -algebra homomorphism (respectively, isomorphism) is a ring homomorphism (respectively, isomorphism) mapping to such that for all and .
It is easy to check that if is an -algebra, then is a ring with identity and has a natural left and right -module structure defined by , where is just the multiplication in the ring . Every ring with identity is actually an -algebra.
Since -algebra has the ring structure and module structure by Definition 1, according to Theorem 2 and Theorem 3, we can obtain Goursat’s lemma for -algebras easily.

Corollary 1. (Goursat’s lemma for -algebras). Let be a commutative ring with identity and be -algebras.

(1)Let be a subalgebra of , and then are subalgebras of such that is an algebraic ideal of for , and the map is an -algebra isomorphism, where .(2)Suppose that are subalgebras of such that is an algebraic ideal of for , and is an -algebra isomorphism, then is a subalgebra of .(3)The construction given in (1) and (2) is inverse to each other.

Example 1. Let , which is an -algebra. We want to find out all the subalgebras of .

Step 1. Find out all the subalgebras ’s () and their corresponding algebraic ideals in . Here, we omit the case when the corresponding algebraic ideal is itself.

Case 1. The subalgebra with 8 elements of , and the algebraic ideals of are as follows:Thus,and has 2 elements, has 4 elements, and has 8 elements.

Case 2. The subalgebra with 4 elements of , and the algebraic ideals of areand has 2 elements for , and has 4 elements.

Case 3. For the subalgebras , their algebraic ideals are only and for .

Step 2. According to Corollary 1, we first construct the isomorphism and then give the subalgebras of .

Case 4. (with 2 elements). Here, we only discuss one case. We have an isomorphism , and the subalgebra of is

Case 5. (with 4 elements).(a)We have an isomorphism defined by , and the subalgebra of is(b)We have an isomorphism which is an identity or defined by , and the subalgebra of for the identity case is and the subalgebra of for the other case is(c)For the isomorphism which is an identity or defined by , the subalgebra of for the identity case is and the subalgebra of for the other case is(d)For the isomorphism which is an identity or defined by , the subalgebra of for the identity case is and the subalgebra of for the other case is

Case 6. (with 8 elements). The isomorphism is an identity, and the subalgebra of is

3. The Zassenhaus Lemma for Groups, Rings, -Modules, and -Algebras

In group theory, it is well known that the four basic isomorphism theorems (Fundamental Homomorphism Theorem, Diamond Isomorphism Theorem, Freshman Theorem, and Correspondence Theorem) about homomorphism and their structure are very useful in the study of groups. All of these theorems have analogues in other algebraic structures, such as rings, vector spaces, -modules, and -algebras. Furthermore, for the Second Isomorphism Theorem, the set-theoretic version of the product formula (i.e., if are subgroups of , then there is a natural bijection between and ), establishes a bijection which is the same as the bijection of the Second Isomorphism Theorem, but without the conditions of normality, and the bijection is purely at the set-theoretic level. In 1934, Zassenhaus gave Zassenhaus lemma which is a generalization of the Second Isomorphism Theorem for groups (see Lemma 5.10 in [26] and Lemma 4.52 in [27]). In this section, we give Zassenhaus lemma for rings, -modules, and -algebras, which is also a generalization of the Second Isomorphism Theorem for rings, -modules, and -algebras, respectively.

Theorem 4. (Zassenhaus lemma for groups, Lemma 4.52 in [27]). Suppose that is a group with subgroups such that and , thenand there is an isomorphism

Remark 1. Zassenhaus lemma for groups implies the Second Isomorphism Theorem for groups, i.e., if and are subgroups of a group with , thenIn fact, we can let and . Thus, according to Theorem 4, we haveIndeed, for any where and , since , we have . Conversely, for any where and , there exists such that , i.e., . It follows that . Thus,Further, it is easy to obtain thatTherefore, (37) can be written as .
Following Lemma 4.52 in [27], we can also set , and and then obtain that .

Example 2. Consider the symmetric group with 24 elements which are as follows:Letthen by applying for Theorem 4. In fact, according to the definitions of , we haveand are subgroups of with . Let , then . Thus, By Theorem 4, we have , which means that .
As a generalization of the Second Isomorphism Theorem for rings, by using Theorem 2, we obtained Zassenhaus lemma for rings stated as follows.

Theorem 5. (Zassenhaus lemma for rings). Let be subrings of a ring such that is an ideal of for , then (respectively, ) is an ideal of (respectively, ), and

Proof. Let , then is a subring of . Indeed, for any , where for , since is also a subring of and is an ideal of for , we havewhich implies that is a subring of . According to Theorem 2, letthen , and is a subring of for clearly. Further, letthen , and is an ideal of for . Following Theorem 2, we have , which shows that

Remark 2. The Zassenhaus lemma for rings implies the second isomorphism theorem for rings, i.e., if is a subring of a ring and an ideal of , thenIn fact, let , , , and , since , , and , we haveAccording to Theorem 5, we have .

For -modules, we know that a module is a mathematical object in which things can be added together commutatively by multiplying coefficients and in which most of the rules of manipulating vectors hold. A module is abstractly very similar to a vector space although in modules coefficients are taken in rings that are much more general algebraic objects than the fields used in vector spaces. -modules can be thought of as generalizations of vector spaces and abelian groups. We will also see that they can be regarded as “representations” of a ring.

For the Second Isomorphism Theorem for -modules, by using Goursat’s lemma for -modules, we have the following generalized Zassenhaus lemma.

Theorem 6. (Zassenhaus lemma for -modules). Let be a commutative ring with identity and an -module. Suppose that are submodules of satisfying , then (respectively, ) is a submodule of (respectively, ), and

Proof. Let ; since is also a submodule of and is a submodule of for , we havefor any , which implies that is a submodule of by the submodule criterion [23]. Following Theorem 3, letas in the proof of Theorem 5, we haveand are submodules of with for . Thus, by Theorem 3, we have

Remark 3. Zassenhaus lemma for modules implies the Second Isomorphism Theorem for modules, i.e., if are submodules of a -module , then . In fact, let , , , and ; since , we haveFollowing Theorem 6, we have .
Similarly, we can easily obtain Zassenhaus lemma for vector spaces, which is also a generalization of the Second Isomorphism Theorem for vector spaces; that is, if is a vector space and are linear subspaces of , then . By Theorem 6, we have the following corollary.

Corollary 2. (Zassenhaus lemma for vector spaces). Let be a vector space and are linear subspaces of satisfying , then (respectively, ) is a subspace of (respectively, ), andFor the Second Isomorphism Theorem for -algebras, by using Corollary 1, we have the following generalized Zassenhaus lemma.

Theorem 7. (Zassenhaus lemma for -algebras). Suppose that is an -algebra and are subalgebras of such that is an algebraic ideal of for , then (respectively, ) is an algebraic ideal of (respectively, ), and

Proof. Since is a subring and a submodule of , it tells that is a subalgebra of . Let ; since is an algebraic ideal of for , then is a subring and a submodule of from Theorem 5 and Theorem 6, which implies that is a subalgebra of . Following Theorem 3, letthen we haveand are subalgebras of such that is an algebraic ideal of for according to Theorem 5 and Theorem 6. Thus, by Theorem 3, we have

Remark 4. Zassenhaus lemma for -algebra implies the Second Isomorphism Theorem for -algebra, i.e., if is an -algebra and are subalgebras of such that is an algebraic ideal of , thenIn fact, let , , , and ; since , we haveFollowing Theorem 7, we have .

Example 3. Let , which is an -algebra, , and , then is an algebraic ideal of for , andAccording to Theorem 7, we have

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The work was supported by the National Natural Science Foundation of China (11701111 and 12031003) and Ministry of Science and Technology of China (CSN-XXZ-20120700649).

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Copyright © 2022 Fanning Meng and Junhui Guo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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