Ebtihal S. Majeid, Raida D. Mahmood
Department of Mathematic, College of Computer Sciences and Mathematic, University of Mosul, Mosul, Iraq.
DOI: 10.4236/oalib.1105626   PDF    HTML   XML   359 Downloads   1,007 Views   Citations

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.

Keywords

GP-Ideal, J-Regular, Reduced Rings, Right Almost J-Injective Rings

Share and Cite:

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 $\oplus$ , $J\left(R\right)$ , $Y\left(R\right)\left(Z\left(R\right)\right)$ , $r\left(a\right)\left(l\left(a\right)\right)$ and $N\left(R\right)$ , respectively.

2. Characterization of Right JGP-Rings

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

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 $R=\left\{\left[\begin{array}{cc}a& b\\ 0& c\end{array}\right]:a,b,c\in Z_2\right\}$ . Then $J\left(R\right)=\left\{\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right],\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]\right\}$ . Clearly $r\left(\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]\right)$ 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 $a\in J\left(R\right)$ and $a+I\in R/I$ . Since R is JGP-ring, then $r\left(a\right)$ is left GP-ideal. Let $x+I\in r\left(a+I\right)$ , $ax\in I$ . Since I is pure ideal. Then there exists $y\in I$ such that $ax=axy,\left(x-xy\right)\in r\left(a\right)$ and $r\left(a\right)$ is GP-ideal. So there exist $w\in r\left(a\right)$ and a positive integer n such that

${\left(x-xy\right)}^n=w{\left(x-xy\right)}^n$

$\begin{array}{l}{x}^n-n{x}^{n-1}xy+n\left(n-1\right)\frac{x^{n-2}{x}^2{y}^2}{2!}+\cdots +{\left(xy\right)}^n\\ =w{x}^n-nw{x}^{n-1}xy+\cdots +w{\left(xy\right)}^n\end{array}$

$x^n-n{x}^{n}y+n\frac{\left(n-1\right)x^n{y}^2}{2!}+\cdots +x^n{y}^n=w{x}^n-nw{x}^{n}y+\cdots +w{x}^n{y}^n$

$\begin{array}{c}{x}^n-w{x}^n=n{x}^{n}y-n\frac{\left(n-1\right)x^n{y}^2}{2!}-\cdots -x^n{y}^n-nw{x}^{n}y\\ +n\frac{\left(n-1\right)w{x}^n{y}^2}{2!}+\cdots +w{x}^n{y}^n\end{array}$

So $\left(x^n-w{x}^n\right)\in I$ , and $x^n+I=w{x}^n+I=\left(w+I\right)\left(x^n+I\right)$ . Therefore $r\left(a+I\right)$ is a left GP-ideal. Hence R/I is JGP-ring.

2.3. Proposition 1

If R is right JGP-ring and $r\left(a\right)\subseteq J\left(R\right)$ for all $a\in J\left(R\right)$ , then $r\left(a\right)$ is nil ideal.

Proof: Let R be JGP-ring, then $r\left(a\right)$ is GP-ideal. For every $b\in r\left(a\right)$ there exist a positive integer n and $x\in r\left(a\right)$ such that $b^n=x{b}^n$ , $\left(1-x\right)b^n=0$ . Since $x\in r\left(a\right)\subseteq J\left(R\right)$ , then $x\in J\left(R\right)$ implies $\left(1-x\right)$ is unit. Then there is $v\in R$ such that $v\left(1-x\right)=1$ , so $v\left(1-x\right)b^n=b^n$ then $b^n=0$ . Therefore $r\left(a\right)$ is nil ideal.

A ring R is called reversible ring [2], if for $a,b\in R$ , $ab=0$ implies $ba=0$ . A ring R is called reduced if $N\left(R\right)=0$ . Clearly, reduced rings are reversible.

2.4. Theorem 2

Let R be a reversible. Then R is right JGP-ring iff $r\left(a\right)+r\left(b^n\right)=R$ for all $a\in J\left(R\right)$ and $b\in r\left(a\right)$ , a positive integer n.

Proof: Let R be JGP-ring, then $r\left(a\right)$ is GP-ideal. For every $b\in r\left(a\right)$ and a positive integer n, considering $r\left(a\right)+r\left(b^n\right)\ne R$ . Then there is a maximal ideal M contain $r\left(a\right)+r\left(b^n\right)$ . Since $r\left(a\right)$ is GP-ideal and $b\in r\left(a\right)$ . Then there exists $c\in r\left(a\right)$ and a positive integer n such that $b^n=c{b}^n$ , implies $\left(1-c\right)\in r\left(b^n\right)\subseteq M$ .

But $c\in r\left(a\right)\subseteq M$ , then $1\in M$ , this contradiction with $M\ne R$ . Therefore $r\left(a\right)+r\left(b^n\right)=R$ . Conversely, let $r\left(a\right)+r\left(b^n\right)=R$ . For all $a\in J\left(R\right)$ and $b\in r\left(a\right)$ , then $x+y=1$ when $x\in r\left(a\right)$ and $y\in r\left(b^n\right)$ multiply by $b^n$ we get $x{b}^n=b^n$ , $r\left(a\right)$ 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 $N\left(R\right)\subseteq J\left(R\right)$ .

3.1. Theorem 3

Let R be JGP and NJ-ring. Then R is reduced if, $l\left(a^n\right)\subseteq r\left(a\right)$ for every $a\in R$ , and positive integer n.

Proof: Consider R not reduced ring, then there is $0\ne a\in J\left(R\right)$ and since R is JGP-ring, then $r\left(a\right)$ is left GP-ideal. Implies $b\in r\left(a\right)$ and a positive integer n such that $a^n=b{a}^n$ , $\left(1-b\right)\in l\left(a^n\right)\subseteq r\left(a\right)$ . So $a=ab$ . Since $b\in r\left(a\right)$ , then $ab=0$ implies $a=0$ and this a contradiction. Therefore R is reduced.

A ring R is called regular if for every $x\in R,x\in xRx$ [4] .

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

3.2. Theorem 4

If $J\left(R\right)=N\left(R\right)$ and $l\left(a^n\right)\subseteq r\left(a\right)$ for all $a\in R$ , 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 $N\left(R\right)=0$ . Since $J\left(R\right)=N\left(R\right)$ , then $J\left(R\right)=0$ . Therefore R is J-regular.

Conversely: it is clear.

3.3. Definition 1

Let $M_R$ be a module with $S=End\left(M_R\right)$ . The module M is called right almost J-injective, if for any $a\in J\left(R\right)$ , there exists an S-sub module $X_a$ of M such that $l_M{r}_R\left(a\right)=Ma\oplus X_a$ 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 $J\left(R\right)\subseteq Y\left(R\right)$ [6] .

From Proposition 2 we get:

3.5. Corollary 1

If R is right almost J-injective and NJ-ring, then $N\left(R\right)\subseteq Y\left(R\right)$ .

An element $a\in R$ is said to be strongly regular if $a=a^{2}b$ for some $b\in R$ [4] .

3.6. Theorem 5

Let R be NJ, JGP and right almost J?injective ring. Then every element in $J\left(R\right)$ is strongly regular. If $l\left(a^n\right)\subseteq r\left(a\right)$ for all $a\in R$ , and positive integer n.

Proof: For all $0\ne a\in J\left(R\right)$ , then $a^2\in J\left(R\right)$ . Since R is almost J-injective ring, then there exist a left ideal X in R such that $Ra\oplus X_a=l\left(r\left(a\right)\right)=l\left(r\left(a^2\right)\right)=R{a}^2\oplus X_a$ , by using Theorem 3, $a\in l\left(r\left(a\right)\right)=l\left(r\left(a^2\right)\right)=R{a}^2\oplus X_a$ . For all $b\in R$ and $x\in X$ , $a=b{a}^2+x$ , then $a^2=ab{a}^2+ax$ implies $a^2-ab{a}^2=ax\in Ra\cap X_a=0$ , $a^2=ab{a}^2$ . Therefore $\left(1-ab\right)\in l\left(a^2\right)\subseteq r\left(a\right)$ . 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.

References

[1]	Mahmood, R.D. (2000) On Pure Ideals and Pure Sub Modules. Ph.D. Thesis, Mosul University, Mosul.
[2]	Cohn, P.M. (1999) Reversible Rings. Bulletin of the London Mathematical Society, 31, 641-648.[CrossRef]
[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.