JGP-Ring

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 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 , $r\left(a\right)$ 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

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

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

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

$\begin{array}{c}{x}^{n}-w{x}^{n}=n{x}^{n}y-n\frac{\left(n-1\right){x}^{n}{y}^{2}}{2!}-\cdots -{x}^{n}{y}^{n}-nw{x}^{n}y\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+n\frac{\left(n-1\right)w{x}^{n}{y}^{2}}{2!}+\cdots +w{x}^{n}{y}^{n}\end{array}$

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 , 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  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 $x\in R,x\in xRx$  .

Following , 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  .

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  .

3.4. Proposition 2

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

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 $b\in R$  .

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.

Cite this paper

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

  Mahmood, R.D. (2000) On Pure Ideals and Pure Sub Modules. Ph.D. Thesis, Mosul University, Mosul.  Cohn, P.M. (1999) Reversible Rings. Bulletin of the London Mathematical Society, 31, 641-648. https://doi.org/10.1112/S0024609399006116  Chang, L. and Soo, Y.P. (2018) When Nilpotents Are Contained in Jacobson Radicals. Journal of the Korean Mathematical Society, 55, 1193-1205.  Rege, M.B. (1986) On Von Neumann Rings and SF-Ring. Mathematica Japonica, 31, 927-936.  Zhao, Y. and Zhou, S.J. (2011) On JPP-Ring, JPF-Rings and J-Regular Rings. Interna-tional Mathematical Forum, 6, 1691-1696.  Mahmood, R.D. (2013) On Almost J-Injectivity and J-Regularity of Rings. Tikrit Journal of Pure Science, 18, 206-210. 