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
,
,
,
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 [1],
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 [2], 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 [3] 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
[4] .
Following [5], a ring R is J-regular if for each
, there exists
such that
. Every regular ring is J-regular ring [5] .
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 [6] .
3.4. Proposition 2
If R is almost J-injective ring, then
[6] .
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
[4] .
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.