Brown-McCoy Radical in Restricted Graded Version

ABSTRACT


A. INTRODUCTION
In this paper, we use ⊲ to express ideal of . Let be a ring and let be a nonzero element of . The element is called a nilpotent element if = 0 for some ∈ ℤ + . If ∀ ∈ , = 0 for some ∈ ℤ + , is called a nil ring. Based on its history, the concept of radical class was observed by Këthe by investigating the property of the class = { |∀ ∈ , ∃ ∈ ℤ + ∋ = 0} of rings. In other words, the class consists of all nil rings (Gardner & Wiegandt, 2004). K̈ethe also discovered some facts on the class of all nil rings. For all rings ∈ then / ∈ , ∀ ⊲ . Furthermore, the largest ideal of a ring which consists of nilpotent elements, is also the member of the class . Finally, if there exists ring and ⊲ ∋ ∈ and / ∈ implies ∈ . These properties of the class of all nil rings motivated Amitsur and Kurosh to define a radical class of rings. A class of rings is said to be radical if / ∈ , ∀0 ≠ proper ideal of , for every ring , ( ) = Σ{ ⊲ | ∈ } ∈ and for every ring , there exist an ideal of and , / ∈ implies ∈ (Gardner & Wiegandt, 2004). Directly, we can infer that the class of all nil rings is a radical class. However, the class 0 = { is a ring | = {0} for some ∈ ℤ + } is not a radical class. In the development of radical theory, there are two types radical based on the construction. The lower radical and the upper radical. The lower radical ℒ 0 of the class 0 is being investigated by Baer, and it is denoted by . Furthermore, since it follows from the fact that the radical class is precisely the upper radical class of the class of all prime rings, the Baer radical class is also called the prime radical class of rings. On the other hand, the class ℒ of all locally nilpotent rings forms a radical class of rings and it is called the Levitzki radical. The structure of Levitzki radical can be seen in (Gardner & Wiegandt, 2004), and the implementation of Levitzki radical in a skew polynomial ring can be seen in (Hong & Kim, 2019).
The nilpotent property of skew generalized power series ring has been discussed in (Ouyang & Liu, 2013). Moreover, a nilpotent derived from the implementation Jacobson radical class = { |( ,∘) forms a group, where ∘ = + − for every , ∈ } of graded group ring is described in (Ilić-Georgijević, 2021). The definition of a graded ring will be given later in Definition 7 in this section. In fact, it follows from (Prasetyo & Melati, 2020) that the Jacobson radical ( ) of a ring is two-sided brace. Some property of nilpotent group and the property of skew left brace of nilpotent type are described in (Smoktunowicz, 2018) and (Cedó et al., 2019).
On the other hand, 0 ≠ ⊲ , is essential if ∩ ≠ {0}, ∀0 ≠ ⊲ and it will be denoted by ⊲∘ . Moreover, is special if consists of prime rings, for every ring ∈ then every nonzero ideal of is also contained in , and for every essential ideal of such that ∈ implies ∈ . An upper radical class ( ) of a special class is special. It follows from the fact that is precisely ( ), where is the class of all prime rings. Hence, is special. Some properties related to special classes of rings and their generalization and their implementation in the development of the radical theory of rings and modules can be seen in (France-Jackson et al., 2015;Prasetyo et al., 2017Prasetyo, Setyaningsih, et al., 2016;Wahyuni et al., 2017).
Furthermore, the class of all simple rings with unity is denoted by ℳ. The upper radical (ℳ) was being observed by Brown and McCoy, and it is called the Brown-McCoy radical class, and it is denoted by (Gardner & Wiegandt, 2004). On the other hand, Emil Ilić-Georgijević in his paper (Ili'c-Georgijevi'c, 2016) introduce a large graded Brown-McCoy radical of a graded ring and compare with the classical graded Brown-McCoy of a graded ring. In this paper, for a fixed group , we scrutinize the restricted −graded Brown-McCoy radical which is denoted by by using fundamental concept of radical class of ring for graded. Moreover for any ring , we explain what ( ) is.
We provide some examples of simple rings with unity and their counterexamples. Example 1. Consider the following concrete simple rings with unity 1. Every field is a simple ring with unity. 2. Let be a field. Then the set ( ) of all matrices of the size × over forms a simple ring with unity We give simple concrete rings which do not contain unity. Example 2.
1. Let ∞ ( ) be the ring of all infinite matrices which are row infinite over a ring , that is, every matrix in ∞ ( ) has a countably infinite number of rows, but almost all entries in each row are equal to 0. In the case, is a field, then ∞ ( ) is simple. Clearly, the center of ∞ ( ) is {0}. Therefore, the simple ring ∞ ( ) does not contain the identity element. 2. The ideal 2ℤ 4 = {0,2} of ℤ 4 = {0,1,2,3} is a simple ring, and it does not have unity.
The Cayley tables of the addition and multiplication modulo 4 of ℤ 4 respectively are shown in Table 1 and Table 2 below.  0  1  2  3  0  0  1  2  3  0  0  0  0  0  1  1  2  3  0  1  0  1  2  3  2  2  3  0  1  2  0  2  0  2  3  3  0  1  2  3  0  3  2  1 We further use the set ℤ 4 of all integer numbers modulo 4 to construct a graded ring in Example 9. In fact, for every special class of rings, is essentially closed. However, every class of prime rings does not necessarily to be essentially closed. We provide the following examples to make these conditions clear.

Example 4.
Let be a prime ring. The prime ring is called a * −ring if / ∈ for every nonzero proper ideal of . The definition of * −ring was introduced by Halina Korolczuk in 1981 (Prasetyo et al., 2017). Let * be the symbol to express the class of all * −rings. However, * is not closed under essential extension.
It follows from Example 4 that there is a class of prime rings, but it is not essentially closed. This condition motivated the existence of the definition of essential cover and essential closure.
In general, the essential closure of a special class of rings is itself since ( ) = for every ∈ {0,1,2, … }. However, in the case of nonspecial class of ring, the essential closure of strictly contains . We provide the following example.

Example 6.
It follows from Example 4 that the class * is not special. The essential closure of * of * strictly contains * .
A graded ring is one of the kinds of rings such that its structure is being investigated by prominent authors. We give highlight some research outcomes related to the existence of graded rings. In the point of view of an epsilon category, the multiplicity of graded algebras and epsilon-strongly groupoid graded can be found in (Das, 2021) and , respectively. The studies on graded ring related to Leavitt path algebra can be seen in (R. Hazrat, 2014;Roozbeh Hazrat et al., 2018;Lännström, 2020;Vaš, 2020aVaš, , 2020b. Furthermore, the studies on graded ring related to the specific structure of rings and modules, namely weakly prime ring, non-commutative rings, prime spectrum, unique factorization rings, positively graded rings, simple rings, −Noetherian ring, and Dedekind rings can be accessed in (Abu-Dawwasb, 2018;Al-Zoubi & Jaradat, 2018;Alshehry & Abu-Dawwasb, 2021;Çeken & Alkan, 2015;Kim & Lim, 2020;Wahyuni et al., 2020;Wijayanti et al., 2020) respectively. Thus, it is interesting to investigate some further properties and structures related to graded rings. On the other hand, the purpose of this research is to determine what ( ) is. Finally, in the following definition, we provide the definition a graded ring.

Definition 7.
Let be a monoid. A ring is a −graded ring if =⊕ ∈ where the set { | ∈ } is the collection of additive subgroups of ∋ ℎ ⊆ ℎ ∀ , ℎ ∈ (Kim & Lim, 2020). In order to make the reader be clear in understanding the concept of a graded ring, we provide the following examples.

Example 9.
Consider the set ℤ 4 = {0,1,2,3} and let be any ring. Now the set 2×2 ( ) is the set of all 2 × 2 matrices over ring . Define 0 = ( 0 0 ) , 1 = ( 0 0 0 0 ) , 2 = ( 0 0 ) , 3 = ( 0 0 0 0 ). We therefore have = 0 ⊕ 1 ⊕ 2 ⊕ 3 and ℎ ⊆ ℎ for every , ℎ ∈ 4 . Thus, we can infer that is a ℤ 4 −graded ring. Some further and nontrivial examples of graded rings can be accessed in (Alshehry & Abu-Dawwasb, 2021) and (Pratibha et al., 2017). Moreover, the monoid can be strictly replaced by any arbitrary group. Let and be graded rings with respect to , the symbol will denote the underlying ungraded ring. A ring homomorphism which maps to is called a graded homomorphism of degree (ℎ, ) if ( ) ⊆ ℎ , ∀ ∈ . The existence of this graded homomorphism motivated the existence of the definition of graded radical. For simply, the definition of graded radical class can be seen in Definition 2 in (Fang & Stewart, 1992), which is similar to the definition of radical class in the graded version.
Furthermore, we shall follow the construction of the restricted graded radical introduced by Hongjin Fang and Patrick Stewart in their paper (Fang & Stewart, 1992). Let be radical, = { | is a −graded ring and ∈ }. Moreover, for further consideration, it will be called a graded radical of . Some properties of graded radical related to the normality and specialty of the graded radical can be seen in (Fang & Stewart, 1992). The previous work on Brown-McCoy has been developed by Emil Ilić-Georgijević in 2016. He introduced a large Brown-McCoy radical for graded ring (Ili'c-Georgijevi'c, 2016). Moreover, the aim of this research is to describe what restricted graded Brown-McCoy radical is by following the work of (Fang & Stewart, 1992) and for every ring , we also describe what ( ) is. The flowchart of this research can be seen in the Figure 1 below.

B. METHODS
This research is conducted using a qualitative method derived from facts and known concepts from a literature study. To gain some properties of radical of rings, we follow some concepts and radical construction (Gardner & Wiegandt, 2004). There are lower radical class and upper radical class. In this part, we focus on the upper radical class construction. We post the known result in the early part of Section C. We start with the structure of what the Brown-McCoy radical of a ring actually is and prove the property completely. In the development of graded radicals, the graded version of the radical class of rings has been being investigated. Gardner and Plant, in their paper (Gardner & Plant, 2009), investigated and compared the Jacobson radical and graded Jacobson radical. The homogeneity radicals defined by the nilpotency of a graded semigroup ring are described in (Hong et al., 2018). Some further fundamental structures of graded radical of rings can also be studied from (Lee & Puczylowski, 2014), (Hong et al., 2014) and (Mazurek et al., 2015). However, we shall follow the concept of restricted graded radical which was introduced by (Fang & Stewart, 1992) to construct the restricted graded Brown-McCoy radical. Finally, in virtue of the construction of the restricted graded Brown-McCoy radical; we can determine the structure of the restricted graded Brown-McCoy radical.

C. RESULT AND DISCUSSION
We separate this part into two subsections. In the first part, we describe what the Brown-McCoy of a ring actually is. Moreover, in the second part, we describe the construction of restricted graded Brown-McCoy radical and give some of its properties.

Radical theory of rings (Initiated by Këthe in 1930)
Fundamental properties and concept related to radical classes of rings were developed and completely described in (Gardner & Wiegandt, 2004) (Fang & Stewart, 1992) developed graded radical class of rings and later (Ili'c-Georgijevi'c, 2016) introduce a large graded Brown-McCoy radical Some implementation of the fundamental properties of radical class were found, for example: construction of brace (Prasetyo & Melati, 2020) In this research, we follow the work of (Fang & Stewart, 1992) to determine what the restricted graded Brown-McCoy is and for any ring , we explain what ( ) is.
In fact, = (ℳ) = { | there is no / ∈ ℳ for every ideal of }y. Now let ∈ ℳ. Suppose and be ideals of such that = {0}. Since the ring is a simple ring, in the case of = implies = {0} or in the case of = implies = {0}. Hence, {0} is a prime ideal of . Thus, is prime. So, we may deduce that ℳ consists of prime rings. Furthermore ℳ is hereditary, and it is essentially closed. This gives ℳ is special. Thus is hereditary. Furthermore, since ℳ is essentially closed, the essential cover ℰℳ of ℳ coincide with ℳ. It follows from Theorem 3.7.2 in (Gardner & Wiegandt, 2004)

Graded Brown-McCoy Radical
Let be any group. It follows from the construction of restricted −graded radical described in (Fang & Stewart, 1992) that a restricted −graded Brown-McCoy radical class can be defined as = { is a −graded ring | ∈ }, where is underlying ungraded of the ring . For further consideration, we call the class of rings by graded Brown-McCoy radical. In the next theorem, we will show that the graded Brown-McCoy radical is hereditary.

Theorem 11.
Let be any group. The restricted −graded Brown-McCoy radical is hereditary.

Proof.
It can be directly inferred by the hereditaries of the Brown-McCoy radical and Proposition 2 in (Fang & Stewart, 1992) that is hereditary. However, we will provide the proof in detail for the reader. Now let ∈ and let be any nonzero homogenous proper ideal of . Then =⊕ ∈ and =⊕ ℎ∈ ∩ ℎ . Hence, is a −graded ring. It is clear that is a ideal of ∈ . Since is hereditary, ∈ . So, we can deduce that ∈ . Thus, is hereditary. ▀ The hereditaries of explained in Theorem 11 implies the following property.

Theorem 12.
Let be any group, ( ) is the intersection of all −graded ideals of such that / is −graded simple ring with unity, where is a −graded ring.

Proof.
Let be any −graded ring. In virtue of Theorem 2, is hereditary. Furthermore, since is hereditary and it follows from Proposition 2 in (Fang & Stewart, 1992) that ( ) = ( ( )) , where ( ( )) =⊕ ∈ { ( ) ∩ | ∈ }. Now let { }, ∈ Λ, where Λ is index, be the collection of all ideals of such that / ∈ ℳ. Define =⊕ ∈ ∩ for every ∈ Λ. It is clear that is a −graded ideal of such that is simple −graded with unity. Now the intersection of all { }, ∈ Λ is On the other hand, ( ) = ( ( )) (2) It follows from equations (1) and (2) that ( ( )) is precisely the intersection of all −graded ideals of such that / is −graded simple ring with unity which completes the proof. ▀ On the other hand, a radical class is called an −radical if it is normal and supernilpotent. The detail of the definition of normal and supernilpotent can be accessed in (Gardner & Wiegandt, 2004). In fact, the Brown-McCoy radical is not an −radical as also explained in (Gardner & Wiegandt, 2004) which implies that the property explained in Theorem 3.18.14 in (Gardner & Wiegandt, 2004) does not hold for a ring of Morita context . However, it is interesting to investigate how about the restricted graded version for the Brown-McCoy radical. If the property also does not hold for the restricted version, then a counter-example should be exist.

D. CONCLUSION AND SUGGESTIONS
It follows from the research outcomes of this research that for a fixed group , we can contruct the restricted graded Brown-McCoy and let be ring, ( ) =∩ { | / is a −graded which is a member of ℳ} . For further research, we can be continue to investigate the restricted −graded for Levitzki radical, Thierrin radical, anti simple radical, and Behrens radical.