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JGP-Ring

DOI: 10.4236/oalib.1105626    37 Downloads   80 Views  

ABSTRACT

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-rings are developed; many of them extend known results.

1. Introduction

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.

2. Characterization of Right JGP-Rings

Call a right JGP-rings, if for every a J ( R ) , r ( a ) is left GP-ideal. Clearly, every left GP-ideal [1], r ( a ) is GP-ideal for every a J ( R ) .

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 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.

2.2. Theorem 1

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.

2.3. Proposition 1

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 [2], if for a , b R , a b = 0 implies b a = 0 . A ring R is called reduced if N ( R ) = 0 . Clearly, reduced rings are reversible.

2.4. Theorem 2

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.

3. JGP-Rings and Other Rings

In this section we consider the connection between JGP-rings and J-regular rings.

Following [3] a ring is called NJ, if N ( R ) J ( R ) .

3.1. Theorem 3

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 [4] .

Following [5], a ring R is J-regular if for each a J ( R ) , there exists x R such that a = a x a . Every regular ring is J-regular ring [5] .

3.2. Theorem 4

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.

3.3. Definition 1

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 [6] .

3.4. Proposition 2

If R is almost J-injective ring, then J ( R ) Y ( R ) [6] .

From Proposition 2 we get:

3.5. Corollary 1

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 [4] .

3.6. Theorem 5

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.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

Cite this paper

Majeid, E. and Mahmood, R. (2019) JGP-Ring. Open Access Library Journal, 6, 1-4. doi: 10.4236/oalib.1105626.

References

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[2] Cohn, P.M. (1999) Reversible Rings. Bulletin of the London Mathematical Society, 31, 641-648. https://doi.org/10.1112/S0024609399006116
[3] Chang, L. and Soo, Y.P. (2018) When Nilpotents Are Contained in Jacobson Radicals. Journal of the Korean Mathematical Society, 55, 1193-1205.
[4] Rege, M.B. (1986) On Von Neumann Rings and SF-Ring. Mathematica Japonica, 31, 927-936.
[5] Zhao, Y. and Zhou, S.J. (2011) On JPP-Ring, JPF-Rings and J-Regular Rings. Interna-tional Mathematical Forum, 6, 1691-1696.
[6] Mahmood, R.D. (2013) On Almost J-Injectivity and J-Regularity of Rings. Tikrit Journal of Pure Science, 18, 206-210.

  
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