The Generalization of Prime Modules

Abstract

Piecewise prime (PWP) module 𝑀_𝑅 is defined in terms of a set of triangulating idempotents in 𝑅. The class of PWP modules properly contains the class of prime modules. Some properties of these modules are investigated here.

1. Introduction

All rings are associative, and 𝑅 denotes a ring with unity 1. The word ideal without the adjective right or left means two-sided ideal. The right annihilator of ideals of 𝑅 is denoted by r.ann_𝑅(𝐼). A ring 𝑅 is 𝑞𝑢𝑎𝑠𝑖-𝐵𝑎𝑒𝑟 (𝐵𝑎𝑒𝑟) if the right annihilator of every right ideal (nonempty subset) of 𝑅 is generated as a right ideal by an idempotent. We now recall a few definitions and results from [1] which motivated our study and serve as the background material for the present work. An idempotent 𝑒∈𝑅 is a left semicentral idempotent if 𝑒𝑥𝑒=𝑥𝑒, for all 𝑥∈𝑅. Similarly right semicentral idempotent can be defined. The set of all left (right) semicentral idempotents of 𝑅 is denoted by 𝑆_𝑙(𝑅)(𝑆_𝑟(𝑅)). An idempotent 𝑒∈𝑅 is semicentral reduced if 𝑆_𝑙(𝑒𝑅𝑒)={0,𝑒}. If 1 is semicentral reduced, then 𝑅 is called semicentral reduced. An ordered set {𝑒₁,…,𝑒_𝑛} of nonzero distinct idempotents of 𝑅 is called a set of left triangulating idempotents of 𝑅 if all the following hold:(i)𝑒₁+⋯+𝑒_𝑛=1,(ii)𝑒₁∈𝑆_𝑙(𝑅),(iii)𝑒_𝑘+1∈𝑆_𝑙(𝑐_𝑘𝑅𝑐_𝑘), where 𝑐_𝑘=1−(𝑒₁+⋯+𝑒_𝑘) for 1≤𝑘≤𝑛.From part (iii) of the previous definition, it can be seen that a set of left triangulating idempotents is a set of pairwise orthogonal idempotents. A set 𝐸={𝑒₁,…,𝑒_𝑛} of left triangulating idempotents of 𝑅 is complete, if each 𝑒_𝑖 is semicentral reduced. A (complete) set of right triangulating idempotents is defined similarly. The cardinalities of complete sets of left triangulating idempotents of 𝑅 are the same and are denoted by 𝜏dim(𝑅) [1, Theorem 2.10]. A ring 𝑅 is called piecewise prime if there exists a complete set of left triangulating idempotents 𝐸={𝑒₁,…,𝑒_𝑛} of 𝑅, such that 𝑥𝑅𝑦=0 implies 𝑥=0 or 𝑦=0 where 𝑥∈𝑒_𝑖𝑅𝑒_𝑗 and 𝑦∈𝑒_𝑗𝑅𝑒_𝑘 for 1≤𝑖,𝑗,𝑘≤𝑛. In view of this definition we say a proper ideal 𝐼 in 𝑅 is a PWP ideal if there is a complete set of left triangulating idempotents 𝐸={𝑒₁,…,𝑒_𝑛}, such that 𝑥𝑅𝑦⊆𝐼 implies 𝑥∈𝐼 or 𝑦∈𝐼, where 𝑥∈𝑒_𝑖𝑅𝑒_𝑗 and 𝑦∈𝑒_𝑗𝑅𝑒_𝑘 for 1≤𝑖,𝑗,𝑘≤𝑛. If 𝑅 is PWP, then it is PWP with respect to any complete set of left triangulating idempotents of 𝑅; furthermore for a ring 𝑅 with finite 𝜏dim(𝑅), 𝑅 is PWP if and only if 𝑅 is quasi-Baer [1, Theorem 4.11].

A nonzero right 𝑅-module 𝑀 is called a prime module if for any nonzero submodule 𝑁 of 𝑀, r.ann_𝑅(𝑁)=r.ann_𝑅(𝑀), and a proper submodule 𝑃 of 𝑀 is a prime submodule of 𝑀 if the quotient module 𝑀/𝑃 is a prime module. The notion of prime submodule was first introduced in [2, 3]; see also [4, 5]. It is easy to see that 𝑀 is a prime 𝑅-module if and only if for any 𝑚∈𝑀, and 𝑏∈𝑅 if 𝑚𝑅𝑏=0, then 𝑚=0 or 𝑀𝑏=0.

In this work the concept of prime modules is developed to piecewise prime modules as it is done for rings in [1]. Throughout this work it is considered that 𝜏dim(𝑅) is finite.

2. Main Results

Definition 1. Let 𝑀 be an 𝑅-module and 𝑆=End_𝑅(𝑀). (1)𝑀 is a piecewise prime (PWP) 𝑅-module with respect to a complete set of left triangulating idempotents 𝐸={𝑒₁,…,𝑒_𝑛} of 𝑅, if for any 𝑚∈𝑀, 𝑒_𝑖∈𝐸, and 𝑏∈𝑅, 𝑚𝑒_𝑖𝑅𝑒_𝑖𝑏=0⇒𝑚𝑒_𝑖=0or𝑀𝑒_𝑖𝑏=0. (1)(2)Let 𝑁 be a submodule of 𝑀. Then 𝑁 is a piecewise prime submodule of 𝑀 with respect to 𝐸 if 𝑀/𝑁 is a PWP module with respect to 𝐸. (3)𝑀 is piecewise endoprime (PWEP) with respect to a complete set of left triangulating idempotents 𝐹={𝑏₁,…,𝑏_𝑚} of 𝑆, such that for each nonzero submodule 𝑁⊆𝑀, 𝑓∈𝑆, and 𝑏_𝑖∈𝐹, if 𝑓𝑏_𝑖𝑁=0, then 𝑓𝑏_𝑖=0.

By Definition 1, 𝑁 is a piecewise prime submodule of 𝑀 with respect to a set of left triangulating idempotents 𝐸 if for any 𝑚∈𝑀, 𝑒_𝑖∈𝐸, and 𝑏∈𝑅, 𝑚𝑒_𝑖𝑅𝑒_𝑖𝑏⊆𝑁⇒𝑚𝑒_𝑖∈𝑁or𝑀𝑒_𝑖𝑏⊆𝑁. (2)

Example 2. Let 𝐸={𝑒₁,…,𝑒_𝑛} be a complete set of left triangulating idempotents o 𝑅. (1)Let 𝑘₁ and 𝑘₂ be two fields and 𝑅=𝑘₁×𝑘₂. Then 𝑀=𝑅/𝑘₁⊕𝑅/𝑘₂ is not a prime module, but it is piecewise prime with respect to {(1,0),(0,1)}. (2)If 𝑀 is a prime 𝑅-module, then it is piecewise prime with respect to any set of left triangulating idempotents of 𝑅. (3)Homomorphic image of 𝑀_𝑅 needs to be PWP with respect to 𝐸. For example, ℤ_ℤ is a PWP module with respect to {0,1}, but ℤ₄ is not PWP because r.ann_𝑟(2)≠r.ann_𝑟(ℤ₄).

Corollary 3. If 𝑀 is a PWP 𝑅-module with respect to 𝐸, then any submodule of 𝑀 is PWP with respect to 𝐸.

Proof. It can be seen by Definition 1.

Proposition 4. Let 𝑅 be a ring with finite triangulating dimension. (1)𝐼 is a PWP ideal of 𝑅 if and only if 𝑅/𝐼 is a PWP 𝑅-module.(2)𝑅 is a PWP ring if and only if 𝑅_𝑅 is PWP.

Proof. The part one is obtained by Definition 1, and for second let 𝐼=0 in part one.

Proposition 5. Let 𝑀 be an 𝑅-module, and let 𝐸={𝑒₁,…,𝑒_𝑛} be a set of left triangulating idempotents of 𝑅. Then the following statements are equivalent:(1)𝑀 is PWP with respect to 𝐸;(2)for each 𝑁⊆𝑀, ideal 𝐼 in 𝑅, and 𝑒_𝑖∈𝐸 if 𝑁𝑒_𝑖𝐼=0 then 𝑁𝑒_𝑖=0 or 𝑀𝑒_𝑖𝐼=0;(3)for each (𝑚)⊆𝑀, ideal (𝑎) in 𝑅, and 𝑒_𝑖∈𝐸 if (𝑚)𝑒_𝑖(𝑎)=0 then (𝑚)𝑒_𝑖=0 or 𝑀𝑒_𝑖(𝑎)=0.

Proof. (1)⇒(2) If 𝑁𝑒_𝑖≠0, then there exists 𝑛∈𝑁, such that 𝑛𝑒_𝑖≠0, and for any 𝑏∈𝐼, 𝑛𝑒_𝑖𝑅𝑒_𝑖𝑏=0. By Definition 1, for each 𝑏∈𝐼, 𝑀𝑒_𝑖𝑏=0. This implies that 𝑀𝑒_𝑖𝐼=0.
(2)⇒(3) In (2), let 𝑁=(𝑚) and 𝐼=(𝑎).
(3)⇒(1) Let 𝑚𝑒_𝑖𝑅𝑒_𝑖𝑏=0 where 𝑚∈𝑀, 𝑒_𝑖∈𝐸, and 𝑏∈𝑅. Thus 𝑚𝑒_𝑖𝑅𝑒_𝑖𝑅𝑒_𝑖𝑏𝑅=0 or (𝑚𝑒_𝑖)𝑒_𝑖(𝑒_𝑖𝑏)=0. By (3), (𝑚𝑒_𝑖)𝑒_𝑖=0 or 𝑀𝑒_𝑖(𝑒_𝑖𝑏)=0. This implies that 𝑚𝑒_𝑖=0 or 𝑀𝑒_𝑖𝑏=0.

Proposition 6. Let 𝑀 be an 𝑅-module, 𝑆=𝐸𝑛𝑑_𝑅(𝑀), let 𝐸={𝑒₁,…,𝑒_𝑛} be a complete set of left triangulating idempotents of 𝑅, and let 𝐹={𝑏₁,…,𝑏_𝑚} be a complete set of left triangulating idempotents of 𝑆. (1)𝑀 is a PWP 𝑅-module with respect to 𝐸 if and only if for each 𝑁⊆𝑀 with 𝑁𝑒_𝑖≠0, 𝑎𝑛𝑛_𝑟(𝑁𝑒_𝑖)=𝑎𝑛𝑛_𝑟(𝑀𝑒_𝑖). (2)If 𝑀_𝑅 is PWP 𝑅-module with respect to 𝐸, then 𝑎𝑛𝑛_𝑟(𝑀) is a PWP ideal of 𝑅 with respect to 𝐸. (3)If 𝑀_𝑅 is PWEP with respect to 𝐹 and retractable, then 𝑎𝑛𝑛_𝑟(𝑀) is a PWP ideal of 𝑅 with respect to 𝐸.

Proof. (1) If 𝑏∈ann_𝑟(𝑁𝑒_𝑖), then there exists 𝑛∈𝑁, such that 𝑛𝑒_𝑖𝑅𝑒_𝑖≠0 and 𝑛𝑒_𝑖𝑅𝑒_𝑖𝑏=0. Since 𝑀 is PWP 𝑅-module with respect to 𝐸 by Definition 1, 𝑀𝑒_𝑖𝑏=0. Hence ann_𝑟(𝑁𝑒_𝑖)=ann_𝑟(𝑀𝑒_𝑖). Conversely let 𝑚𝑒_𝑖𝑅𝑒_𝑖𝑏=0 where 𝑒_𝑖∈𝐸, 𝑚∈𝑀, 𝑏∈𝑅, and 𝑚𝑒_𝑖≠0. Thus 𝑏∈ann_𝑟((𝑚𝑒_𝑖𝑅)𝑒_𝑖) which means 𝑏∈ann_𝑟(𝑀𝑒_𝑖) or 𝑀𝑒_𝑖𝑏=0.
(2) Let 𝐼𝑒_𝑖𝐽⊆ann_𝑟(𝑀) and 𝐼𝑒_𝑖⊆a̸nn_𝑟(𝑀). Since (𝑀𝐼𝑒_𝑖)𝑒_𝑖𝐽=0, and 𝑀 is a PWP 𝑅-module with respect to 𝐸, by Proposition 5, 𝑀𝑒_𝑖𝐽=0. Thus 𝑒_𝑖𝐽⊆ann_𝑟(𝑀). This implies that ann_𝑟(𝑀) is a PWP ideal of 𝑅 with respect to 𝐸.
(3) Let 𝐼𝑒_𝑖𝐽⊆ann_𝑟(𝑀) where 𝐼𝑒_𝑖,𝑒_𝑖𝐽⊆a̸nn_𝑟(𝑀). Since 𝑀 is retractable, then there exists a nonzero homomorphism 𝑓∶𝑀→𝑀𝐼𝑒_𝑖. There exists 𝑏_𝑗∈𝐹, such that 𝑓𝑏_𝑗≠0. Since 𝐼𝑒_𝑖𝐽⊆ann_𝑟(𝑀), 𝑓𝑏_𝑗𝑀𝑒_𝑖𝐽=0. By assumption 𝑀 is PWEP with respect to 𝐹. This implies that 𝑓𝑏_𝑗=0 which is a contradiction. Hence ann_𝑟(𝑀) is a PWP ideal of 𝑅 with respect to 𝐸.

A module 𝑀_𝑅 is called retractable if for any nonzero submodule 𝑁 of 𝑀, Hom_𝑅(𝑀,𝑁)≠0.

Theorem 7. Let 𝑀 be an 𝑅-module, 𝑆=𝐸𝑛𝑑_𝑅(𝑀), and let 𝐹={𝑏₁,…,𝑏_𝑚} be a complete set of left triangulating idempotents of 𝑆. (1)If _𝑆𝑀 is a PWP module with respect to 𝐹, then 𝑆 is a PWP ring. The converse is true when 𝑀_𝑅 is retractable. (2)_𝑆𝑀 is a PWP module with respect to 𝐹, if and only if 𝑀_𝑅 is PWEP with respect to 𝐹.

Proof. (1) Let 𝑓𝑏_𝑖𝑆𝑏_𝑖𝑔=0 where 𝑓,𝑔∈𝑆, 𝑏_𝑖∈𝐹 and 𝑏_𝑖𝑔≠0. Thus there exists 𝑚∈𝑀, such that 𝑏_𝑖𝑔𝑚≠0 and 𝑓𝑏_𝑖𝑆𝑏_𝑖𝑔𝑚=0. Since _𝑆𝑀 is PWP with respect to 𝐹, 𝑓𝑏_𝑖𝑀=0 which means 𝑓𝑏_𝑖=0. Conversely let 𝑓𝑏_𝑖𝑆𝑏_𝑖𝑚=0 and 𝑏_𝑖𝑚≠0. Since 𝑀_𝑅 is retractable, there exists a nonzero homomorphism 𝑏_𝑖𝑔∈Hom_𝑅(𝑀,𝑏_𝑖𝑚𝑅). Thus 𝑓𝑏_𝑖𝑆𝑏_𝑖𝑔=0. Since 𝑆 is PWP, 𝑓𝑏_𝑖=0.
(2) Assume 𝑀 is a PWP  𝑆-module with respect to 𝐹. Let 𝑁⊆𝑀 and 𝑓𝑏_𝑖𝑁=0 where 𝑓∈𝑆 and 𝑏_𝑖∈𝐹. Since _𝑆𝑀 is PWP, by Proposition 6(1), 𝑓𝑏_𝑖𝑀=0. Thus 𝑓𝑏_𝑖=0. Conversely assume 𝑀_𝑅 be PWEP with respect to 𝐹. Let 𝑓𝑏_𝑖𝑆𝑏_𝑖𝑚=0 where 𝑓∈𝑆, 𝑏_𝑖∈𝐹, 𝑚∈𝑀, and 𝑏_𝑖𝑚≠0. If 𝑁=𝑆𝑏_𝑖𝑚, then 𝑓𝑏_𝑖𝑁=0. This implies that 𝑓𝑏_𝑖=0 or 𝑓𝑏_𝑖𝑀=0. Hence _𝑆𝑀 is PWP with respect to 𝐹.

Let 𝑀 be a right 𝑅-module with 𝑆=End_𝑅(𝑀). Then 𝑀_𝑅 is called a quasi-Baer module, if for any 𝑁⊆_𝑆𝑀, l.ann_𝑆(𝑁)=𝑆𝑒, where 𝑒=𝑒²∈𝑆 [6].

Corollary 8. Let 𝑀 be a retractable 𝑅-module, 𝑆=𝐸𝑛𝑑_𝑅(𝑀), and let 𝐹={𝑏₁,…,𝑏_𝑚} be a complete set of left triangulating idempotents of 𝑆. Then the following statements are equivalent:(1)𝑀_𝑅 is a PWEP module with respect to 𝐹;(2)_𝑆𝑀 is a PWP module with respect to 𝐹;(3)𝑀_𝑅 is quasi-Baer.

Proof. (1)⇔(2) This is evident by Theorem 7(2).
(2)⇔(3) By [6, Proposition 4.7], 𝑀_𝑅 is quasi-Baer if and only if 𝑆 is quasi-Baer. By [1, Theorem 4.11], 𝑆 is PWP with respect to 𝐹 if and only if 𝑆 is quasi-Baer. The result is obtained by Theorem 7(1).

Proposition 9. Let Λ be an index set, and let 𝐸={𝑒₁,…,𝑒_𝑛} be a complete set of left triangulating idempotents of 𝑅. (1)Let 𝑀=⊕_𝜆∈Λ𝑀_𝜆. 𝑀 is PWP with respect to 𝐸 if and only if for each 𝜆∈Λ, 𝑀_𝜆 is PWP with respect to 𝐸. (2)Let 𝑀=∏_𝜆∈Λ𝑀_𝜆. 𝑀 is PWP with respect to 𝐸 if and only if for each 𝜆∈Λ, 𝑀_𝜆 is PWP with respect to 𝐸.

Proof. (1) Assume 𝑀 is PWP with respect to 𝐸. If 𝑚_𝜆𝑒_𝑖𝑅𝑒_𝑖𝑏=0, where 𝑚_𝜆∈𝑀_𝜆, 𝑒_𝑖∈𝐸, and 𝑏∈𝑅 then (0,…,𝑚_𝜆,0,…,0)𝑒_𝑖𝑅𝑒_𝑖𝑏=0. Since 𝑀 is PWP, (0,…,𝑚_𝜆,0,…,0)𝑒_𝑖=0 or 𝑀𝑒_𝑖𝑏=0. This implies that 𝑚_𝜆𝑒_𝑖=0 or 𝑀_𝜆𝑒_𝑖𝑏=0 which means for each 𝜆∈Λ, 𝑀_𝜆 is PWP with respect to 𝐸. Conversely assume that for each 𝜆∈Λ, 𝑀_𝜆 is PWP with respect to 𝐸, and (𝑚₁,…,𝑚_𝑛,0,…)𝑒_𝑖𝑅𝑒_𝑖𝑏=0. This implies that 𝑚_𝜆𝑒_𝑖𝑅𝑒_𝑖𝑏=0. Since 𝑀_𝜆 is PWP with respect to 𝐸, 𝑚_𝜆𝑒_𝑖=0 or 𝑀𝑒_𝑖𝑏=0. Hence (𝑚₁,…,𝑚_𝑛,0,…)𝑒_𝑖=0 or 𝑀𝑒_𝑖𝑏=0. Thus 𝑀 is PWP with respect to 𝐸.
(2) It can be seen by similar method as in part (1).

Corollary 10. Let 𝐸={𝑒₁,…,𝑒_𝑛} be a complete set of left triangulating idempotents of 𝑅, let 𝑀 be an 𝑅-module, and let 𝐹 be a free 𝑅-module. (1)𝑅 is quasi-Baer if and only if 𝐹 is a PWP module with respect to 𝐸. (2)𝑀 is PWP with respect to 𝐸 if and only if   𝐹⊗_𝑅  M is PWP with respect to 𝐸.

Proof. It follows by [1, Theorem 4.11] and Proposition 9.

Proposition 11. Let 𝑀 be an 𝑅-module, and 𝑆=𝐸𝑛𝑑_𝑅(𝑀). Then _𝑆𝑀 is prime if and only if 𝜏𝑑𝑖𝑚(𝑆)=1, and 𝑀_𝑅 is quasi-Baer.

Proof. (⇒) Since 𝑀 is a prime 𝑆-module, then for each 𝑁⊆𝑀, l.ann_𝑆(𝑁)=l.ann_𝑆(𝑀)=0. This implies that 𝑀_𝑅 is quasi-Baer. If 𝑒²=𝑒∈𝑆, then 𝑀=𝑒𝑀⊕(1−𝑒)𝑀. Since _𝑆𝑀 is prime, l.ann_𝑆(𝑒𝑀)=l.ann_𝑆((1−𝑒)𝑀)=l.ann_𝑆(𝑀). This implies that 𝑒=1 or 𝑒=0. Thus 𝜏dim(𝑆)=1.
(⇐) Let 𝑁 be any submodule of _𝑆𝑀. Since 𝑀_𝑅 is quasi-Baer, l.ann_𝑆(𝑁)=𝑆𝑒, where 𝑒∈𝑆_𝑟(𝑆). Since 𝜏dim(𝑆)=1, 𝑒∈{0,1}. If 𝑒=1, then 𝑁=0. Thus 𝑒=0. This implies that for each nonzero submodule 𝑁⊆_𝑆𝑀, l.ann_𝑆(𝑁)=l.ann_𝑆(𝑀)=0. This means _𝑆𝑀 is prime.

It is folklore that prime radical plays an important role in the study of rings [7]. Following this concept is developed for modules of course by using a complete set of left triangulating idempotents of 𝑅.

Definition 12. Let 𝑀 be an 𝑅-module, let 𝑁 be a proper submodule of 𝑀, and let 𝐸={𝑒₁,…,𝑒_𝑛} be a complete set of left triangulating idempotents of 𝑅. (1)The piecewise prime radical of 𝑁 in 𝑀 with respect to 𝐸 is denoted by PRad(𝑁) and is defined to be the intersection of all piecewise prime submodules of 𝑀 with respect to 𝐸 containing 𝑁. (2)PRad(𝑀) means the intersection of all piecewise prime submodules of 𝑀 with respect to 𝐸. If 𝑀 has no piecewise prime submodule with respect to 𝐸, then PRad(𝑀)=𝑀.

Proposition 13. Let 𝑁 be a submodule of 𝑅-module 𝑀. (1)If 𝑁 is a submodule of 𝑅-module 𝑀, then 𝑃𝑅𝑎𝑑(𝑁)⊆𝑃𝑅𝑎𝑑(𝑀).(2)If 𝑃𝑅𝑎𝑑(𝑀)=𝐾, then 𝑃𝑅𝑎𝑑(𝑀/𝐾)=0. (3)If 𝑀=⊕_𝑖∈𝐼𝑀_𝑖 is a direct sum of submodules 𝑀_𝑖, thenPRad(𝑀)=_𝑖∈𝐼PRad𝑀_𝑖. (3)

Proof. Let 𝐸={𝑒₁,…,𝑒_𝑛} be a complete set of left triangulating idempotents of 𝑅.(1)Let 𝐾 be any piecewise prime submodule of 𝑀 with respect to 𝐸. If 𝑁⊆𝐾, then PRad(𝑁)⊆𝐾. If 𝑁⊈𝐾, then by the definition it is easy to see that 𝑁∩𝐾 is a piecewise prime submodule of 𝑁 with respect to 𝐸. Thus PRad(𝑁)⊆(𝐾∩𝑁)⊆𝐾. Hence PRad(𝑁)⊆PRad(𝑀).(2)Let 𝑃/𝐾 be a piecewise prime submodules of 𝑀/𝐾 with respect to 𝐸. By definition (𝑀/𝐾)/(𝑃/𝐾) is a piecewise prime module with respect to 𝐸. Thus 𝑀/𝑃 is a a piecewise prime module with respect to 𝐸. This implies that 𝑃 is a piecewise prime submodules of 𝑀 with respect to 𝐸. Hence PRad(𝑀/𝐾)=0.(3)By (1) for each 𝑖∈𝐼, PRad(𝑀_𝑖)⊆PRad(𝑀). This implies that PRad𝑀_{𝑖∈𝐼𝑖}⊆PRad(𝑀). (4)Let (𝑚_𝑖)_𝑖∈𝐼∈𝑀⧵⊕_𝑖∈𝐼PRad(𝑀_𝑖). Then there exists 𝑖∈𝐼, such that 𝑚_𝑖∉PRad(𝑀_𝑖). By the definition there exists a piecewise prime submodule 𝑁_𝑖⊆𝑀_𝑖 with respect to 𝐸, such that 𝑚_𝑖∉𝑁_𝑖. If 𝐾=𝑁_𝑖⊕(⊕_𝑖≠𝑗𝑀_𝑗), then 𝐾 is a piecewise prime submodule of 𝑀 with respect to 𝐸, and 𝑚∉𝐾. Thus 𝑚∉PRad(𝑀). It means that PRad(𝑀)=_𝑖∈𝐼PRad𝑀_𝑖. (5)