A Ring R is called right JGP-ring; if for every a ∈ J (R), r (a) is a left GP-ideal. In this paper, we first introduced and characterize JGP-ring, which is a proper generalization of right GP-ideal. Next, various properties of right JGP-r ings are developed; many of them extend known results.
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 ⊕ , J ( R ) , Y ( R ) ( Z ( R ) ) , r ( a ) ( l ( a ) ) and N ( R ) , respectively.
Call a right JGP-rings, if for every a ∈ J ( R ) , r ( a ) is left GP-ideal. Clearly, every left GP-ideal [
1) The ring Z of integers is right JGP-ring which is not every ideal of Z is GP-ideal.
2) Let R = { [ a b 0 c ] : a , b , c ∈ Z 2 } . Then J ( R ) = { [ 0 0 0 0 ] , [ 0 1 0 0 ] } . Clearly r ( [ 0 1 0 0 ] ) is left GP-ideal. Therefore R is JGP-ring.
Let R be a right JGP-ring and I is pure ideal. Then R/I is JGP-ring.
Proof: Let a ∈ J ( R ) and a + I ∈ R / I . Since R is JGP-ring, then r ( a ) is left GP-ideal. Let x + I ∈ r ( a + I ) , a x ∈ I . Since I is pure ideal. Then there exists y ∈ I such that a x = a x y , ( x − x y ) ∈ r ( a ) and r ( a ) is GP-ideal. So there exist w ∈ r ( a ) and a positive integer n such that
( x − x y ) n = w ( x − x y ) n
x n − n x n − 1 x y + n ( n − 1 ) x n − 2 x 2 y 2 2 ! + ⋯ + ( x y ) n = w x n − n w x n − 1 x y + ⋯ + w ( x y ) n
x n − n x n y + n ( n − 1 ) x n y 2 2 ! + ⋯ + x n y n = w x n − n w x n y + ⋯ + w x n y n
x n − w x n = n x n y − n ( n − 1 ) x n y 2 2 ! − ⋯ − x n y n − n w x n y + n ( n − 1 ) w x n y 2 2 ! + ⋯ + w x n y n
So ( x n − w x n ) ∈ I , and x n + I = w x n + I = ( w + I ) ( x n + I ) . Therefore r ( a + I ) is a left GP-ideal. Hence R/I is JGP-ring.
If R is right JGP-ring and r ( a ) ⊆ J ( R ) for all a ∈ J ( R ) , then r ( a ) is nil ideal.
Proof: Let R be JGP-ring, then r ( a ) is GP-ideal. For every b ∈ r ( a ) there exist a positive integer n and x ∈ r ( a ) such that b n = x b n , ( 1 − x ) b n = 0 . Since x ∈ r ( a ) ⊆ J ( R ) , then x ∈ J ( R ) implies ( 1 − x ) is unit. Then there is v ∈ R such that v ( 1 − x ) = 1 , so v ( 1 − x ) b n = b n then b n = 0 . Therefore r ( a ) is nil ideal.
A ring R is called reversible ring [
Let R be a reversible. Then R is right JGP-ring iff r ( a ) + r ( b n ) = R for all a ∈ J ( R ) and b ∈ r ( a ) , a positive integer n.
Proof: Let R be JGP-ring, then r ( a ) is GP-ideal. For every b ∈ r ( a ) and a positive integer n, considering r ( a ) + r ( b n ) ≠ R . Then there is a maximal ideal M contain r ( a ) + r ( b n ) . Since r ( a ) is GP-ideal and b ∈ r ( a ) . Then there exists c ∈ r ( a ) and a positive integer n such that b n = c b n , implies ( 1 − c ) ∈ r ( b n ) ⊆ M .
But c ∈ r ( a ) ⊆ M , then 1 ∈ M , this contradiction with M ≠ R . Therefore r ( a ) + r ( b n ) = R . Conversely, let r ( a ) + r ( b n ) = R . For all a ∈ J ( R ) and b ∈ r ( a ) , then x + y = 1 when x ∈ r ( a ) and y ∈ r ( b n ) multiply by b n we get x b n = b n , r ( a ) is GP-ideal. Therefore R is JGP-ring.
In this section we consider the connection between JGP-rings and J-regular rings.
Following [
Let R be JGP and NJ-ring. Then R is reduced if, l ( a n ) ⊆ r ( a ) for every a ∈ R , and positive integer n.
Proof: Consider R not reduced ring, then there is 0 ≠ a ∈ J ( R ) and since R is JGP-ring, then r ( a ) is left GP-ideal. Implies b ∈ r ( a ) and a positive integer n such that a n = b a n , ( 1 − b ) ∈ l ( a n ) ⊆ r ( a ) . So a = a b . Since b ∈ r ( a ) , then a b = 0 implies a = 0 and this a contradiction. Therefore R is reduced.
A ring R is called regular if for every x ∈ R , x ∈ x R x [
Following [
If J ( R ) = N ( R ) and l ( a n ) ⊆ r ( a ) for all a ∈ R , 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 N ( R ) = 0 . Since J ( R ) = N ( R ) , then J ( R ) = 0 . Therefore R is J-regular.
Conversely: it is clear.
Let M R be a module with S = E n d ( M R ) . The module M is called right almost J-injective, if for any a ∈ J ( R ) , there exists an S-sub module X a of M such that l M r R ( a ) = M a ⊕ X a as left S-module. If R R is almost J-injective, then we call R is a right almost J-injective ring [
If R is almost J-injective ring, then J ( R ) ⊆ Y ( R ) [
From Proposition 2 we get:
If R is right almost J-injective and NJ-ring, then N ( R ) ⊆ Y ( R ) .
An element a ∈ R is said to be strongly regular if a = a 2 b for some b ∈ R [
Let R be NJ, JGP and right almost J?injective ring. Then every element in J ( R ) is strongly regular. If l ( a n ) ⊆ r ( a ) for all a ∈ R , and positive integer n.
Proof: For all 0 ≠ a ∈ J ( R ) , then a 2 ∈ J ( R ) . Since R is almost J-injective ring, then there exist a left ideal X in R such that R a ⊕ X a = l ( r ( a ) ) = l ( r ( a 2 ) ) = R a 2 ⊕ X a , by using Theorem 3, a ∈ l ( r ( a ) ) = l ( r ( a 2 ) ) = R a 2 ⊕ X a . For all b ∈ R and x ∈ X , a = b a 2 + x , then a 2 = a b a 2 + a x implies a 2 − a b a 2 = a x ∈ R a ∩ X a = 0 , a 2 = a b a 2 . Therefore ( 1 − a b ) ∈ l ( a 2 ) ⊆ r ( a ) . Since R is reduced, then a = a 2 b . Therefore a is strongly regular element.
The authors declare no conflicts of interest regarding the publication of this paper.
Majeid, E.S. and Mahmood, R.D. (2019) JGP-Ring. Open Access Library Journal, 6: e5626. https://doi.org/10.4236/oalib.1105626