Throughout this paper, every ring is an associative ring with identity unless otherwise stated. Let R be a ring, the direct sum, the Jacobson radical, the right (left) singular, the right (left) annihilator and the set of all nilpotent elements of R are denoted by , , , and , respectively.
2. Characterization of Right JGP-Rings
Call a right JGP-rings, if for every , is left GP-ideal. Clearly, every left GP-ideal , is GP-ideal for every .
2.1. Example 1
1) The ring Z of integers is right JGP-ring which is not every ideal of Z is GP-ideal.
2) Let . Then . Clearly is left GP-ideal. Therefore R is JGP-ring.
2.2. Theorem 1
Let R be a right JGP-ring and I is pure ideal. Then R/I is JGP-ring.
Proof: Let and . Since R is JGP-ring, then is left GP-ideal. Let , . Since I is pure ideal. Then there exists such that and is GP-ideal. So there exist and a positive integer n such that
So , and . Therefore is a left GP-ideal. Hence R/I is JGP-ring.
2.3. Proposition 1
If R is right JGP-ring and for all , then is nil ideal.
Proof: Let R be JGP-ring, then is GP-ideal. For every there exist a positive integer n and such that , . Since , then implies is unit. Then there is such that , so then . Therefore is nil ideal.
A ring R is called reversible ring , if for , implies . A ring R is called reduced if . Clearly, reduced rings are reversible.
2.4. Theorem 2
Let R be a reversible. Then R is right JGP-ring iff for all and , a positive integer n.
Proof: Let R be JGP-ring, then is GP-ideal. For every and a positive integer n, considering . Then there is a maximal ideal M contain . Since is GP-ideal and . Then there exists and a positive integer n such that , implies .
But , then , this contradiction with . Therefore . Conversely, let . For all and , then when and multiply by we get , is GP-ideal. Therefore R is JGP-ring.
3. JGP-Rings and Other Rings
In this section we consider the connection between JGP-rings and J-regular rings.
Following  a ring is called NJ, if .
3.1. Theorem 3
Let R be JGP and NJ-ring. Then R is reduced if, for every , and positive integer n.
Proof: Consider R not reduced ring, then there is and since R is JGP-ring, then is left GP-ideal. Implies and a positive integer n such that , . So . Since , then implies and this a contradiction. Therefore R is reduced.
A ring R is called regular if for every  .
Following , a ring R is J-regular if for each , there exists such that . Every regular ring is J-regular ring  .
3.2. Theorem 4
If and for all , and positive integer n, then R is JGP-ring iff R is J-regular ring.
Proof: Let R be JGP-ring, from Theorem 3 R is reduced ring implies that . Since , then . Therefore R is J-regular.
Conversely: it is clear.
3.3. Definition 1
Let be a module with . The module M is called right almost J-injective, if for any , there exists an S-sub module of M such that as left S-module. If is almost J-injective, then we call R is a right almost J-injective ring  .
3.4. Proposition 2
If R is almost J-injective ring, then  .
From Proposition 2 we get:
3.5. Corollary 1
If R is right almost J-injective and NJ-ring, then .
An element is said to be strongly regular if for some  .
3.6. Theorem 5
Let R be NJ, JGP and right almost J?injective ring. Then every element in is strongly regular. If for all , and positive integer n.
Proof: For all , then . Since R is almost J-injective ring, then there exist a left ideal X in R such that , by using Theorem 3, . For all and , , then implies , . Therefore . Since R is reduced, then . Therefore a is strongly regular element.