Graded Weakly Strongly Quasi-Primary Ideals over Commutative Graded Rings

Abstract

In this article, we introduce and examine the concept of graded weakly strongly quasi primary ideals. A proper graded ideal P of R is said to be a graded weakly strongly quasi primary (shortly, Gwsq-primary) ideal if whenever $0\ne xy\in P$, for some homogeneous elements $x,y\in R$, then $x^2\in P$ or $y^n\in P$, for some positive integer n. Many examples and properties of Gwsq-primary ideals are given. Among several results, we compare Gwsq-primary ideals and other classical graded ideals such as graded strongly quasi primary ideals, graded weakly primary ideals and graded weakly 2-prime ideals etc.

1. Preliminaries and Introduction

Let G be a group and R be a commutative ring with nonzero unity 1. Then R is called G-graded if ${\displaystyle R=\underset{g\in G}{⨁}{R}_g}$ with $R_g{R}_h\subseteq R_{gh}$ for all $g,h\in G$, where $R_g$ is an additive subgroup of R for all $g\in G$, here $R_g{R}_h$ denotes the additive subgroup of R consisting of all finite sums of elements $a_g{b}_h$ with $a_g\in R_g$ and $b_h\in R_h$. We denote this by $G\left(R\right)$. The elements of $R_g$ are called homogeneous of degree g. If $a\in R$, then a can be written uniquely as ${\displaystyle a=\sum _{g\in G}{a}_g}$, where $a_g$ is the component of a in $R_g$ and $a_g=0$ except for finitely many. The additive subgroup $R_e$ is in fact a subring of R and $1\in R_e$. The set of all homogeneous elements of R is ${\displaystyle \bigcup _{g\in G}{R}_g}$ and is denoted by $h\left(R\right)$. A graded ring R is said to be graded reduced if R has no nonzero homogeneous nilpotent elements. Let P be an ideal of a G-graded ring R. Then P is called a graded ideal if ${\displaystyle P=\underset{g\in G}{⨁}(P\cap R_g)}$, i.e., for $a\in P$, ${\displaystyle a=\sum _{g\in G}{a}_g}$ where $a_g\in P$ for all $g\in G$. An ideal of a graded ring is not necessarily a graded ideal. For more terminology, see [1,2].

The concept of graded prime ideals and its generalizations have a distinguished place in graded commutative algebra since not only they are used in characterizing certain class of graded rings, but also they have some applications to other areas such as general topology, algebraic geometry, graph theory etc. (for examples, see [3,4]). Recall from [3] ([4]) that a proper graded ideal P of R is said to be a graded prime (graded weakly prime) ideal if whenever $xy\in P$ ($0\ne xy\in P$), for some $x,y\in h\left(R\right)$, then either $x\in P$ or $y\in P$. Clearly, if P is a prime ideal of R and P is a graded ideal of R, then P is a graded prime ideal of R. On the other hand, the next example shows that a graded prime ideal of R is not necessarily a prime ideal:

Example 1.

Consider $R=\mathbb{Z}\left[i\right]$ and $G={\mathbb{Z}}_2$. Then R is G-graded by $R_0=\mathbb{Z}$ and $R_1=i\mathbb{Z}$. Consider the graded ideal $I=pR$ of R, where p is a prime with $p=c^2+d^2$, for some $c,d\in \mathbb{Z}$. Let $xy\in I$, for some $x,y\in h\left(R\right)$.

Case (1): Assume that $x,y\in R_0$. In this case, $x,y\in \mathbb{Z}$ such that p divides $xy$, and then either p divides x or p divides y, which implies that $x\in I$ or $y\in I$.

Case (2): Assume that $x,y\in R_1$. In this case, $x=ia$ and $y=ib$, for some $a,b\in \mathbb{Z}$ such that p divides $xy=-ab$, and then p divides a or p divides b in $\mathbb{Z}$, which implies that p divides $x=ia$ or p divides $y=ib$ in R. Therefore, $x\in I$ or $y\in I$.

Case (3): Assume that $x\in R_0$ and $y\in R_1$. In this case, $x\in \mathbb{Z}$ and $y=ib$, for some $b\in \mathbb{Z}$ such that p divides $xy=ixb$ in R, that is $ixb=p(\alpha +i\beta )$, for some $\alpha ,\beta \in \mathbb{Z}$. Then we obtain $xb=p\beta$, that is p divides $xb$ in $\mathbb{Z}$, and again p divides x or p divides b, which implies that p divides x or p divides $y=ib$ in R. Thus, $x\in I$ or $y\in I$.

Hence, I is a graded prime ideal of R. On the other hand, I is not a prime ideal of R, as $(c-id)(c+id)\in I$, $(c-id)\notin I$, and $c+id\notin I$.

In addition, recall from [5] ([6]) that a proper graded ideal P of R is said to be a graded primary (graded weakly primary) ideal if whenever $xy\in P$ ($0\ne xy\in P$), for some $x,y\in h\left(R\right)$, then either $x\in P$ or $y\in Grad\left(P\right)$, where $Grad\left(P\right)$ is the graded radical of P, and is defined to be the set of all $r\in R$ such that for each $g\in G$, there exists a positive integer $n_g$ satisfies $r_g^{n_g}\in P$. One can see that $Grad\left(P\right)$ is a graded ideal of R, and if $r\in h\left(R\right)$, then $r\in Grad\left(P\right)$ if and only if $r^n\in P$ for some positive integer n. In 2021, Bataineh and Abu-Dawwas in [7] introduced the concept of graded 2-prime ideals. A proper graded ideal P of R is said to be a graded 2-prime ideal if whenever $xy\in P$, for some $x,y\in h\left(R\right)$, then $x^2\in P$ or $y^2\in P$. Recently, a proper graded ideal P of R is said to be a graded weakly 2-prime ideal if whenever $0\ne xy\in P$, for some $x,y\in h\left(R\right)$, then $x^2\in P$ or $y^2\in P$. In 2022, Abdullah et al. in [8] defined graded strongly quasi primary ideals which is a generalization of graded 2-prime ideals. A proper graded ideal P of R is said to be a graded strongly quasi primary ideal if whenever $xy\in P$, for some $x,y\in h\left(R\right)$, then $x^2\in P$ or $y\in Grad\left(P\right)$. Our aim in this article is following [9] to introduce and study graded weakly strongly quasi primary ideals. In [9], a proper ideal P of a ring R is said to be a weakly strongly quasi primary ideal if whenever $0\ne xy\in P$, for some $x,y\in R$, then either $x^2\in P$ or $y^n\in P$, for some positive integer n. We propose this concept in the graded sense as follows: A proper graded ideal P of a graded ring R is said to be a graded weakly strongly quasi primary (shortly, Gwsq-primary) ideal if whenever $0\ne xy\in P$, for some $x,y\in h\left(R\right)$, then $x^2\in P$ or $y\in Grad\left(P\right)$. Clearly, if R is a graded ring and P is a proper graded ideal of R such that P is a weakly strongly quasi primary ideal, then P is a Gwsq-primary ideal. However, we show that a Gwsq-primary ideal is not necessarily weakly strongly quasi primary. Many examples and properties of Gwsq-primary ideals are given. Among several results, we compare Gwsq-primary ideals and other classical graded ideals such as graded strongly quasi primary ideals, graded weakly primary ideals and graded weakly 2-prime ideals etc.

Research Plan: This research investigates graded weakly strongly quasi-primary ideals, a subclass of quasi-primary ideals in graded rings. Understanding these ideals can provide insights into the structure of graded rings and their applications in algebraic geometry and commutative algebra. The objectives of this paper are as follows: first, to define and characterize graded weakly strongly quasi-primary ideals; second, to explore properties and relationships between these ideals and other classes of graded ideals; and third, to develop results on the behavior of graded weakly strongly quasi-primary ideals in various graded rings. Our research questions are as follows: What are the key properties that distinguish graded weakly strongly quasi-primary ideals from other graded ideals? How do these properties affect the structure of graded rings? What are the implications of these ideals for applications in algebraic geometry? The methodology includes performing a theoretical study using algebraic methods and ring theory, creating formal definitions, and establishing crucial theorems about graded weakly strongly quasi-primary ideals. Specific examples and counterexamples are used to demonstrate the qualities and limitations of these ideals. Additionally, prospective applications in algebraic geometry and other domains are investigated where possible.

2. Main Prospects of Discussing Gwsq-Primary Ideals over Commutative Graded Rings in Practical Applications

The concept of Gwsq-primary ideals in commutative graded rings is a sophisticated branch of abstract algebra that is especially important in algebraic geometry and commutative algebra. Below, we provide an overview of its potential practical uses and significant topics of discussion.

In algebraic geometry, graded rings are frequently equivalent to homogeneous coordinate rings of projective varieties; see [10]. Gwsq-primary ideals can help us to grasp sheaves on these types, notably in terms of sections and cohomologies. The features of Gwsq-primary ideals can influence the structure and decomposition of modules over graded rings, which is important in both algebraic and computational contexts. In computational algebra, understanding the nature of these graded ideals may lead to more efficient methods for computing Gröbner bases in graded contexts. This can be used to solve systems of polynomial equations. Graded rings are frequently utilized in algorithms linked to symbolic computation. The unique features of Gwsq-primary ideals may offer optimizations in algorithms working with these rings. In the broader context of commutative algebra, researching Gwsq-primary ideals can aid in understanding the decomposition of ideals in graded rings, particularly when finding basic components; for more details, see [11]. Graded rings appear in advanced fields of mathematical physics such as string theory. The study of Gwsq-primary ideals may connect with these disciplines, particularly the algebraic structures that underpin physical theories; for more terminology, see [12]. In conclusion, potential topics for discussion include the investigation of algebraic structures in geometry, combinatorial applications, homological features, and even computational methodologies. While abstract, these concepts can have far-reaching ramifications for mathematics and its applications in a variety of disciplines; for more, we refer interested readers to [13].

3. Main Results

In this section, we introduce and examine graded weakly strongly quasi primary ideals.

Definition 1.

Let R be a graded ring and let P be a proper graded ideal of R. Then, P is said to be a graded weakly strongly quasi-primary (shortly, Gwsq-primary) ideal if $x^2\in P$ or $y\in Grad\left(P\right)$ whenever $0\ne xy\in P$ for some $x,y\in h\left(R\right)$.

Clearly, if R is a graded ring and P is a proper graded ideal of R such that P is a weakly strongly quasi primary ideal, then P is a Gwsq-primary ideal. However, the next example shows that a Gwsq-primary ideal is not necessarily weakly strongly quasi primary:

Example 2.

Consider $R=\mathbb{Z}\left[i\right]$ and $G={\mathbb{Z}}_2$. Then R is G-graded by $R_0=\mathbb{Z}$ and $R_1=i\mathbb{Z}$. Consider the graded ideal $P=5R$ of R. Then per Example 1, P is a graded prime ideal of R, and hence P is a Gwsq-primary ideal of R. On the other hand, P is not weakly strongly quasi primary since $1-2i,1+2i\in R$ with $0\ne (1-2i)(1+2i)\in P$, ${(1-2i)}^2\notin P$ and ${(1+2i)}^n\notin P$, for all positive integer n.

Proposition 1.

Let R be a graded ring and P be a proper graded ideal of R.

1.

If P is a graded strongly quasi primary ideal of R, then P is a Gwsq-primary ideal of R.

2.

If P is a graded weakly primary ideal of R, then P is a Gwsq-primary ideal of R.

3.

If P is a graded weakly 2-prime ideal of R, then P is a Gwsq-primary ideal of R.

4.

If P is a Gwsq-primary ideal of R and R is graded reduced, then $Grad\left(P\right)$ is a graded weakly prime ideal of R.

5.

If $Grad\left(P\right)$ is a graded weakly prime ideal of R and ${\left(Grad\left(P\right)\right)}^2\subseteq P$, then P is a graded weakly 2-prime ideal of R.

6.

If $Grad\left(P\right)$ is a graded weakly prime ideal of R and ${\left(Grad\left(P\right)\right)}^2\subseteq P$, then P is a Gwsq-primary ideal of R.

Proof. 

• Let $x,y\in h\left(R\right)$ with $0\ne xy\in P$. Then as P is graded strongly quasi primary, either $x^2\in P$ or $y\in Grad\left(P\right)$. Hence, P is a Gwsq-primary ideal of R.
• Let $x,y\in h\left(R\right)$ with $0\ne xy\in P$. Then as P is graded weakly primary, either $x\in P$ or $y\in Grad\left(P\right)$, and clearly if $x\in P$, then $x^2\in P$. Hence, P is a Gwsq-primary ideal of R.
• Let $x,y\in h\left(R\right)$ with $0\ne xy\in P$. Then as P is graded weakly 2-prime, either $x^2\in P$ or $y^2\in P$, and clearly if $y^2\in P$, then $y\in Grad\left(P\right)$. Hence, P is a Gwsq-primary ideal of R.
• Let $x,y\in h\left(R\right)$ with $0\ne xy\in Grad\left(P\right)$. Then there exists a positive integer n such that $x^n{y}^n={\left(xy\right)}^n\in P$, and as R is graded reduced, $0\ne {\left(xy\right)}^n=x^n{y}^n$, and then as P is Gwsq-primary, either $x^{2n}\in P$ or $y^n\in Grad\left(P\right)$, which implies that either $x^{2n}\in P$ or $y^{kn}\in P$, for some positive integer k, which means that either $x\in Grad\left(P\right)$ or $y\in Grad\left(P\right)$. Hence, $Grad\left(P\right)$ is a graded weakly prime ideal of R.
• Let $x,y\in h\left(R\right)$ with $0\ne xy\in P$. Then $0\ne xy\in Grad\left(P\right)$, and then as $Grad\left(P\right)$ is graded weakly prime, either $x\in Grad\left(P\right)$ or $y\in Grad\left(P\right)$, which implies that either $x^2\in {\left(Grad\left(P\right)\right)}^2\subseteq P$ or $y^2\in {\left(Grad\left(P\right)\right)}^2\subseteq P$. Hence, P is a graded weakly 2-prime ideal of R.
• The result holds from (3) and (5).

□

The following examples demonstrate how fundamentally different the Gwsq-primary ideals notion is from other graded ideals concepts.

Example 3.

Let $R={\mathbb{Z}}_{12}\left[i\right]$ and $G={\mathbb{Z}}_2$. Then R is G-graded by $R={\mathbb{Z}}_{12}$ and $R_1=i{\mathbb{Z}}_{12}$. Consider the graded ideal $P=\left\{0\right\}$ of R. Then P is a Gwsq-primary ideal of R. On the other hand, P is not graded strongly quasi primary since $3,4\in h\left(R\right)$ with $3.4\in P$, $3^2\notin P$ and $4\notin Grad\left(P\right)$.

Example 4.

Let $R=K[X,Y]$, where K is a field, and $G=\mathbb{Z}$. Then R is G-graded by ${\displaystyle R_n=\underset{i+j=n,i,j\ge 0}{⨁}K{X}^i{Y}^j}$, for all non-negative integer n, $R_n=0$, otherwise. Consider the graded ideal $P=⟨X^3,XY,Y^3⟩$ of R. Then, per ([8], Example 2.5), P is a graded strongly quasi primary ideal of R, and so Gwsq-primary. On the other hand, P is not graded weakly 2-prime since $X,Y\in h\left(R\right)$ with $0\ne XY\in P$, $X^2\notin P$ and $Y^2\notin P$.

Example 5.

Consider $R=K[X,Y]$, where K is a field, and $G=\mathbb{Z}$. Then R is G-graded by ${\displaystyle R_n=\underset{i+j=n,i,j\ge 0}{⨁}K{X}^i{Y}^j}$, for all non-negative integer n, $R_n=0$, otherwise. Consider the graded ideal $I=⟨X^2,XY⟩$ of R. Then I is not a graded weakly prime ideal since $X,Y\in h\left(R\right)$ with $0\ne XY\in I$ but $X\notin I$ and $Y\notin I$. Also, note that $Grad\left(I\right)=⟨X⟩$ is a graded prime ideal of R. We show that I is a graded weakly 2-prime ideal of R. Assume that $f(X,Y),g(X,Y)\in h\left(R\right)$ such that $0\ne f(X,Y)g(X,Y)\in I$. Then $f(X,Y)g(X,Y)\in Grad\left(I\right)$, and then either X divides $f(X,Y)$ or X divides $g(X,Y)$, which implies that either $X^2$ divides $f^2(X,Y)$ or $X^2$ divides $g^2(X,Y)$, that is either $f^2(X,Y)\in I$ or $g^2(X,Y)\in I$. Hence, I is a graded weakly 2-prime ideal of R, and hence I is a Gwsq-primary ideal of R.

Example 6.

Let $R=K[X,Y]$, where K is a field, and $G=\mathbb{Z}$. Then R is G-graded by ${\displaystyle R_n=\underset{i+j=n,i,j\ge 0}{⨁}K{X}^i{Y}^j}$, for all non-negative integer n, $R_n=0$, otherwise. Consider the graded ideal $I=⟨X^2⟩$ of R. Then $T=R/I$ is G-graded by $T_n=(R_n+I)/I$, for all $n\in \mathbb{Z}$. Consider the graded ideal $P=⟨\overline{XY},{\overline{X}}^2⟩$ of T, where $\overline{\alpha }=\alpha +I$, for all $\alpha \in R$. Then P is a Gwsq-primary ideal of T; to see this, let $\overline{f},\overline{g}\in h\left(T\right)$ with $\overline{0}\ne \overline{fg}\in P$. Then X divides $fg$, and then either X divides f or X divides g, which implies that either $X^2$ divides $f^2$ or $X^2$ divides $g^2$, which means that either ${\overline{f}}^2\in P$ or ${\overline{g}}^2\in P$. Thus, P is a graded weakly 2-prime ideal of T, and hence Gwsq-primary. On the other hand, P is not graded weakly primary since $\overline{X},\overline{Y}\in h\left(T\right)$ with $\overline{0}\ne \overline{XY}\in P$, $\overline{X}\notin P$ and $\overline{Y}\notin Grad\left(P\right)$.

Definition 2.

Let R be a graded ring, $g\in G$ and P be a graded ideal of R with $P_g\ne R_g$. Then:

1.

P is said to be a g-strongly quasi primary ideal if whenever $xy\in P$, for some $x,y\in R_g$, then $x^2\in P$ or $y\in Grad\left(P\right)$.

2.

P is said to be a g-weakly strongly quasi primary (shortly, g-wsq-primary) ideal if whenever $0\ne xy\in P$, for some $x,y\in R_g$, then $x^2\in P$ or $y\in Grad\left(P\right)$.

Clearly, every g-strongly quasi-primary ideal is g-wsq-primary. However, the graded ideal $\left\{0\right\}$ is always g-wsq-primary, but is not necessarily g-strongly quasi-primary ideal.

Proposition 2.

Let R be a graded ring and let P be a g-wsq-primary ideal of R for some $g\in G$. If P is not g-strongly quasi-primary, then $P_g^2=\left\{0\right\}$.

Proof. 

Supposing that $P_g^2\ne \left\{0\right\}$, we show that P is g-strongly quasi primary. Let $xy\in P$, for some $x,y\in R_g$ such that $x^2\notin P$. If $xy\ne 0$, then $y\in Grad\left(P\right)$. Suppose that $xy=0$. If $x{P}_g\ne \left\{0\right\}$, then there is $q\in P_g$ such that $xq\ne 0$, so $0\ne x(y+q)=xq\in P$, then $y+q\in Grad\left(P\right)$, and then $y\in Grad\left(P\right)$. If $y{P}_g\ne \left\{0\right\}$, then there is $p\in P_g$ such that $yp\ne 0$, so $0\ne (x+p)y=yp\in P$, since ${(x+p)}^2\notin P$, and then $y\in Grad\left(P\right)$. Assume that $x{P}_g=y{P}_g=\left\{0\right\}$. Since $P_g^2\ne \left\{0\right\}$, there exist $c,d\in P_g$ such that $cd\ne 0$. Then $0\ne (x+c)(y+d)=cd\in P$, since ${(x+c)}^2\notin P$, then $y+d\in Grad\left(P\right)$, and so $y\in Grad\left(P\right)$. Thus, we conclude that P is g-strongly quasi primary ideal of R. □

The next example shows that a graded ideal P of R with $P_g^2=\left\{0\right\}$ for some $g\in G$ is not necessarily a g-wsq-primary ideal of R. In addition, the next example shows that a proper graded ideal P of R with $P^2=\left\{0\right\}$ is not necessarily a Gwsq-primary ideal of R.

Example 7.

Let $R={\mathbb{Z}}_{12}\left[i\right]$ and $G={\mathbb{Z}}_2$. Then R is G-graded by $R={\mathbb{Z}}_{12}$ and $R_1=i{\mathbb{Z}}_{12}$. Consider the graded ideal $P=6R$ of R. Then $P^2=\left\{0\right\}$ and $P_0^2=\left\{0\right\}$. However, P is not a 0-wsq-primary ideal of R. Indeed, we have $2,3\in R_0$ with $0\ne 2.3\in P$, but $2^2\notin P$ and $3^n\notin P$, for every positive integer n. Clearly, P is not a Gwsq-primary ideal of R.

Corollary 1.

Let R be a reduced graded ring and let P be a g-wsq-primary ideal of R for some $g\in G$. If P is not g-strongly quasi-primary, then $P_g=\left\{0\right\}$.

Proposition 3.

Let R be a graded ring.

1.

If P is a Gwsq-primary ideal of R and $P=Grad\left(P\right)$, then P is a graded weakly prime ideal of R.

2.

If P is a graded weakly primary ideal of R and Q a graded ideal of R containing P, then $PQ$ is a Gwsq-primary ideal of R.

3.

If P is a graded weakly primary ideal of R, then $P^2$ is a Gwsq-primary ideal of R.

Proof. 

• Let $0\ne xy\in P$, for some $x,y\in h\left(R\right)$. Since P is a Gwsq-primary ideal of R, then $x^2\in P$ or $y\in Grad\left(P\right)$. Hence, $x\in P$ or $y\in P$, and so P is a graded weakly prime ideal of R.
• Let $0\ne xy\in PQ\subseteq P$, for some $x,y\in h\left(R\right)$. Then $x\in P$ or $y\in Grad\left(P\right)$. Since $P\subseteq Q$, we get $x^2\in PQ$ or $y\in Grad\left(P\right)=Grad\left(PQ\right)$. Hence, $PQ$ is a Gwsq-primary ideal of R.
• It follows from (2).

□

Theorem 1.

Let P be a proper graded ideal of a graded ring R. Then P is a Gwsq-primary ideal of R if and only if for all $a\in h\left(R\right)$, either $Ra\subseteq (P:a)$ or $(P:a)\subseteq Grad\left(P\right)$ or $(P:a)\subseteq Ann\left(a\right)$.

Proof. 

Suppose that P is a Gwsq-primary ideal of R. Take $a\in h\left(R\right)$. If $a^2\in P$, then it is clear that $Ra\subseteq (P:a)$. Assume that $a^2\notin P$. Let $b\in (P:a)$, that is, $ab\in P$. If $0\ne ab$, then either $a^2\in P$ or $b^n\in P$, for some positive integer n. Since $a^2\notin P$, we conclude $b\in Grad\left(P\right)$. If $0=ab$, then $b\in Ann\left(a\right)$, i.e., $(P:a)\subseteq Ann\left(a\right)\bigcup Grad\left(P\right)$. This implies that $(P:a)\subseteq Grad\left(P\right)$ or $(P:a)\subseteq Ann\left(a\right)$. Conversely, choose $x,y\in h\left(R\right)$ such that $0\ne xy\in P$ and $x^2\notin P$. Then by assumption, $(P:x)\subseteq Grad\left(P\right)$ or $(P:x)\subseteq Ann\left(x\right)$. If $(P:x)\subseteq Grad\left(P\right)$, then $y\in (P:x)\subseteq Grad\left(P\right)$, which is desired. Let $(P:x)\subseteq Ann\left(x\right)$. This means $xy=0$, which is a contradiction. □

Theorem 2.

Let R be a graded ring and P be a g-wsq-primary ideal of R, for some $g\in G$. Then whenever $\left\{0\right\}\ne x{I}_g\subseteq P$, for some $x\in R_g$ and a graded ideal I of R, we have $x^2\in P$ or $I_g\subseteq Grad\left(P\right)$.

Proof. 

Let $x^2\notin P$ and $y\in I_g$. If $0\ne xy\in P$, then we have $y\in Grad\left(P\right)$. Now, Assume that $xy=0$. Choose $z\in I_g$ such that $xz\ne 0$. Since $xz\in P$, we have $z\in Grad\left(P\right)$. On the other hand, $0\ne x(z+y)\in P$. This implies $z+y\in Grad\left(P\right)$ and thus $y\in Grad\left(P\right)$. Hence, $I_g\subseteq Grad\left(P\right)$. □

A commutative graded ring R with unity is said to be a graded domain (graded field) if R has no homogeneous zero divisors (every nonzero homogeneous element of R is unit). Clearly, if R is a domain (field) and R is graded, then R is a graded domain (graded field). However, a graded domain (graded field) is not necessarily domain (field) ([14], Example 2.4).

Theorem 3.

Let R be a graded ring.

1.

If every Gwsq-primary ideal of R is graded prime, then R is a graded domain.

2.

If R is a graded field, then every Gwsq-primary ideal of R is graded prime.

Proof. 

• Because $\left\{0\right\}$ is a Gwsq-primary ideal of R, $\left\{0\right\}$ is a graded prime ideal of R, and then R is a graded domain.
• Because R is a graded field, $\left\{0\right\}$ is the only Gwsq-primary ideal of R and it is a graded prime ideal of R.

□

Proposition 4.

Let P and Q be two Gwsq-primary ideals of a graded ring R. If $Grad\left(P\right)=Grad\left(Q\right)$, then $P\bigcap Q$ is a Gwsq-primary ideal of R.

Proof. 

Let $x,y\in h\left(R\right)$ such that $0\ne xy\in P\bigcap Q$. Then $0\ne xy\in P$ and $0\ne xy\in Q$. Suppose that $y\notin Grad(P\bigcap Q)=Grad\left(P\right)\bigcap Grad\left(Q\right)=Grad\left(P\right)=Grad\left(Q\right)$. Since P is Gwsq-primary, $x^2\in P$. Similarly, $x^2\in Q$. So, $x^2\in P\bigcap Q$, and hence $P\bigcap Q$ is a Gwsq-primary ideal of R. □

Corollary 2.

Let ${\left\{P_i\right\}}_{i\in ▵}$ be a family of Gwsq-primary ideals of a graded ring R. If $Grad\left(P_i\right)=Grad\left(P_j\right)$, for all $i,j\in ▵$, then ${\displaystyle \bigcap _{i\in ▵}{P}_i}$ is a Gwsq-primary ideal of R.

Let R and S be two G-graded rings. Then, a ring homomorphism $f:R\to S$ is said to be a graded ring homomorphism if $f\left(R_g\right)\subseteq S_g$ for all $g\in G$ [1].

Proposition 5.

Let $f:R\to S$ be a graded ring homomorphism.

1.

If f is an epimorphism and P is a Gwsq-primary ideal of R containing $Ker\left(f\right)$, then $f\left(P\right)$ is a Gwsq-primary ideal of S.

2.

If f is a monomorphism and Q is a Gwsq-primary ideal of S, then $f^{-1}\left(Q\right)$ is a Gwsq-primary ideal of R.

Proof. 

• Let $s,t\in h\left(S\right)$ with $0\ne st\in f\left(P\right)$. Then, there exist $a,b\in h\left(R\right)$ such that $s=f\left(a\right)$, $t=f\left(b\right)$, and $0\ne f\left(ab\right)=st\in f\left(P\right)$. Because $Ker\left(f\right)\subseteq P$, we have $0\ne ab\in P$. This implies that $a^2\in P$ or $b\in Grad\left(P\right)$, which means that $f{\left(a\right)}^2=f\left(a^2\right)=s^2\in f\left(P\right)$ or $t\in f\left(Grad\right(P\left)\right)=Grad\left(f\right(P\left)\right)$. Thus, $f\left(P\right)$ is a Gwsq-primary ideal of S.
• Let $x,y\in h\left(R\right)$ such that $0\ne xy\in f^{-1}\left(Q\right)$. Because $Ker\left(f\right)=\left\{0\right\}$, we obtain $0\ne f\left(xy\right)=f\left(x\right)f\left(y\right)\in Q$. Hence, we have $f{\left(x\right)}^2=f\left(x^2\right)\in Q$ or $f\left(y\right)\in Grad\left(Q\right)$; thus, $x^2\in f^{-1}\left(Q\right)$ or $y\in f^{-1}\left(Grad\left(Q\right)\right)=Grad\left(f^{-1}\left(Q\right)\right)$. We can conclude that $f^{-1}\left(Q\right)$ is a Gwsq-primary ideal of R.

□

Let R be a G-graded ring and P be a graded ideal of R. Then $R/P$ is a G-graded ring by ${(R/P)}_g=(R_g+P)/P$, for all $g\in G$ ([1]).

Proposition 6.

Let $P\subseteq Q$ be proper graded ideals of a graded ring R.

1.

If Q is a Gwsq-primary ideal of R, then $Q/P$ is a Gwsq-primary ideal of $R/P$.

2.

If $Q/P$ is a Gwsq-primary ideal of $R/P$ and P is a Gwsq-primary ideal of R, then Q is a Gwsq-primary ideal of R.

Proof. 

• Define $f:R\to R/P$ by $f\left(r\right)=r+P$, for all $r\in R$. Then f is a graded ring epimorphism with $Ker\left(f\right)=P\subseteq Q$, and then per Proposition 5, $f\left(Q\right)=Q/P$ is a Gwsq-primary ideal of $R/P$.
• Let $0\ne xy\in Q$, for some $x,y\in h\left(R\right)$. If $0\ne xy\in P$, then $x^2\in P\subseteq Q$ or $y\in Grad\left(P\right)\subseteq Grad\left(Q\right)$. If $xy\notin P$, then we have $0+P\ne xy+P=(x+P)(y+P)\in Q/P$, and so $x^2+P={(x+P)}^2\in Q/P$ or $y+P\in Grad(Q/P)=Grad\left(Q\right)/P$. It means that $x^2\in Q$ or $y\in Grad\left(Q\right)$. Thus, Q is a Gwsq-primary ideal of R.

□

Proposition 7.

If P is a Gwsq-primary ideal of a graded ring R and S is a graded subring of R with $S⊈P$, then $S\bigcap P$ is a Gwsq-primary ideal of S.

Proof. 

Define $f:S\to R$ by $f\left(s\right)=s$, for all $s\in S$. Then f is a graded ring monomorphism, and then per Proposition 5, $f^{-1}\left(P\right)=S\bigcap P$ is a Gwsq-primary ideal of S. □

Corollary 3.

If P is a Gwsq-primary ideal of a graded ring R and $R_e⊈P$, then $P_e$ is a wsq-primary ideal of $R_e$.

Proof. 

Apply Proposition 7 on $S=R_e$. □

Let $R_1$ and $R_2$ be two G-graded rings. Then $R=R_1\times R_2$ is a G-graded ring by $R_g={\left(R_1\right)}_g\times {\left(R_2\right)}_g$, for all $g\in G$ [1]. Also, a G-graded ring R is said to be a cross product if $R_g$ contains a unit element for all $g\in G$ [1].

Theorem 4.

Let $R_1$ and $R_2$ be two G-graded rings such that $R_1$ and $R_2$ are crossed products. Assume that $R=R_1\times R_2$ and $\left\{0\right\}\ne P=P_1\times P_2$, where $P_1$ and $P_2$ are graded ideals of $R_1$ and $R_2$, respectively. If P is a Gwsq-primary ideal of R, then either $P_1=R_1$ and $P_2$ is a Gwsq-primary ideal of $R_2$, or $P_2=R_2$ and $P_1$ is a Gwsq-primary ideal of $R_1$.

Proof. 

Without loss of generality, we may assume that $P_1\ne \left\{0\right\}$. Choose $0\ne a\in P_1$; then, $a_g\ne 0$ for some $g\in G$, where $a_g\in P_1$ as $P_1$ is a graded ideal. Because $R_2$ is a crossed product, there exists a unit element $r\in {\left(R_2\right)}_g$. Note that $(0,0)\ne (a_g,r)(1,0)\in P$. Because P is a Gwsq-primary ideal of R, we have ${(a_g,r)}^2=(a_g^2,r^2)\in P$ or $(1,0)\in Grad\left(P\right)=Grad\left(P_1\right)\times Grad\left(P_2\right)$. Thus, $r^2\in P_2$ or $1\in P_1$, that is, $P_1=R_1$ or $P_2=R_2$. Suppose that $P_1\ne R_1$, and let $0\ne xy\in P_1$ for some $x,y\in h\left(R_1\right)$; then, $(0,0)\ne (xy,0)=(x,0)(y,0)\in P$ and either ${(x,0)}^2=(x^2,0)\in P$ or $(y,0)\in Grad\left(P\right)$, that is, either $x^2\in P_1$ or $y\in Grad\left(P_1\right)$. Hence, $P_1$ is a Gwsq-primary ideal of $R_1$. In other cases it is similarly possible to show that $P_2$ is a Gwsq-primary ideal of $R_2$. □

Theorem 5.

Let $R_1$ and $R_2$ be two G-graded domains, $R=R_1\times R_2$, $P_1$ and $P_2$ are graded ideals of $R_1$ and $R_2$ respectively.

1.

If $P_1$ is a Gwsq-primary ideal of $R_1$, then $P=P_1\times R_2$ is a Gwsq-primary ideal of R.

2.

If $P_2$ is a Gwsq-primary ideal of $R_2$, then $P=R_1\times P_2$ is a Gwsq-primary ideal of R.

Proof. 

• Let $(0,0)\ne (a,b)(x,y)=(ax,by)\in P$ for some $(a,b),(x,y)\in h\left(R\right)$. If $ax=0$, then either $a=0$ or $x=0$; in turn, either $a^2=0\in P_1$ or $x=0\in Grad\left(P_1\right)$, which implies that either ${(a,b)}^2=(0,b^2)\in P$ or $(x,y)=(0,y)\in Grad\left(P\right)$. Suppose that $ax\ne 0$; then, $0\ne ax\in P_1$, and either $a^2\in P_1$ or $x\in Grad\left(P_1\right)$, which implies that either ${(a,b)}^2=(a^2,b^2)\in P$ or $(x,y)\in Grad\left(P\right)$. Thus, $P=P_1\times R_2$ is a Gwsq-primary ideal of R.
• Similar to (1).

□

Assume that M is an R-module. Then M is said to be G-graded if ${\displaystyle M=\underset{g\in G}{⨁}{M}_g}$ with $R_g{M}_h\subseteq M_{gh}$, for all $g,h\in G$, where $M_g$ is an additive subgroup of M, for all $g\in G$. The elements of $M_g$ are called homogeneous of degree g. It is clear that $M_g$ is an $R_e$-submodule of M, for all $g\in G$. We Assume that ${\displaystyle h\left(M\right)=\bigcup _{g\in G}{M}_g}$. Let N be an R-submodule of a graded R-module M. Then N is said to be a graded R-submodule if ${\displaystyle N=\underset{g\in G}{⨁}(N\bigcap M_g)}$, i.e., for $x\in N$, ${\displaystyle x=\sum _{g\in G}{x}_g}$, where $x_g\in N$, for all $g\in G$. It is known that an R-submodule of a graded R-module is not necessarily graded. Let M be an R-module. The idealization $R(+)M=\left\{\right(r,m):r\in R,m\in M\}$ of M is a commutative ring with component-wise addition and multiplication; $(x,m_1)+(y,m_2)=(x+y,m_1+m_2)$ and $(x,m_1)(y,m_2)=(xy,x{m}_2+y{m}_1)$, for each $x,y\in R$ and $m_1,m_2\in M$. Let G be an Abelian group and M be a G-graded R-module. Then $X=R(+)M$ is G-graded by $X_g=R_g(+)M_g$, for all $g\in G$ [15]. The authors in [15] determined the certain classes of graded ideals such as graded maximal ideal, graded prime ideals, graded primary ideals, graded quasi primary ideals, graded 2-absorbing ideals and graded 2-absorbing quasi primary ideals of graded idealization $R(+)M$. Also, the authors in [8] investigated the graded strongly quasi primary ideals in $R(+)M$. Now, we investigate the Gwsq-primary ideals in $R(+)M$.

Proposition 8

([15], Proposition 3.3). Let G be an Abelian group, M be a G-graded R-module, P be an ideal of R and N be an R submodule of M such that $PM\subseteq N$. Then $P(+)N$ is a graded ideal of $R(+)M$ if and only if P is a graded ideal of R and N is a graded R-submodule of M.

In addition, ([15], Corollary 3.5) characterizes a graded radical of $P(+)N$ as follows:

$$Grad\left(P\right(+\left)N\right)=Grad\left(P\right)(+)M$$

Proposition 9.

Let G be an Abelian group, R a G-graded ring, M a G-graded R-module, and P a graded ideal of R. Then, $P(+)M$ is a Gwsq-primary ideal of $R(+)M$ if and only if P is a Gwsq-primary ideal of R and if $x,y\in An{n}_R\left(M\right)$ whenever $x,y\in h\left(R\right)$ with $xy=0$, $x^2\notin P$, and $y\notin Grad\left(P\right)$.

Proof. 

Suppose that $P(+)M$ is a Gwsq-primary ideal of $R(+)M$. Let $0\ne xy\in P$, for some $x,y\in h\left(R\right)$. Then $(0,0)\ne (x,0)(y,0)\in P(+)M$. Hence, ${(x,0)}^2=(x^2,0)\in P(+)M$ or $(y,0)\in Grad\left(P\right(+\left)M\right)=Grad\left(P\right)(+)M$. Therefore, $x^2\in P$ or $y\in Grad\left(P\right)$, and so P is a Gwsq-primary ideal of R. Now, Assume that $x,y\in h\left(R\right)$ with $xy=0$, $x^2\notin P$ and $y\notin Grad\left(P\right)$. If $x\notin An{n}_R\left(M\right)$, then there exists $m\in M$ such that $xm\ne 0$, and then there exists $g\in G$ such that $x{m}_g\ne 0$. So, $(0,0)\ne (x,0)(y,m_g)\in P(+)M$, and hence ${(x,0)}^2=(x^2,0)\in P(+)M$ or $(y,m_g)\in Grad\left(P(+)M\right)=Grad\left(P\right)(+)M$, which implies that $x^2\in P$ or $y\in Grad\left(P\right)$, which is a contradiction. Thus, $x\in An{n}_R\left(M\right)$. Similarly, $y\in An{n}_R\left(M\right)$. Conversely, let $(0,0)\ne (x,m)(y,s)\in P(+)M$, for some $(x,m),(y,s)\in h\left(R\right(+\left)M\right)$. Then $xy\in P$. If $xy\ne 0$, then we have $x^2\in P$ or $y\in Grad\left(P\right)$, and then ${(x,m)}^2\in P(+)M$ or $(y,s)\in Grad\left(P\right)(+)M=Grad\left(P\right(+\left)M\right)$. Hence, $P(+)M$ is a Gwsq-primary ideal of $R(+)M$. Now, Assume that $xy=0$. If $x^2\notin P$ and $y\notin Grad\left(P\right)$, then $x,y\in An{n}_R\left(M\right)$, and then $(x,m)(y,s)=(0,0)$, which is a contradiction. □

In [1], a graded R-module M is said to be graded reduced if $xm=0$ whenever $x^{2}m=0$ for some $x\in h\left(R\right)$ and $m\in h\left(M\right)$.

Proposition 10.

Let G be an Abelian group, R a G-graded ring, and M a G-graded reduced R-module. Suppose that P is a graded ideal of R and N is a graded R-submodule of M such that $Grad\left(P\right)M\subseteq N$. Then, $P(+)N$ is a Gwsq-primary ideal of $R(+)M$ if and only if P is a Gwsq-primary ideal of R and if $x,y\in An{n}_R\left(N\right)$ whenever $x,y\in h\left(R\right)$, $xy=0$, $x^2\notin P$, and $y\notin Grad\left(P\right)$.

Proof. 

Suppose that $P(+)N$ is a Gwsq-primary ideal of $R(+)M$. Then by applying similar argument as in the proof of Proposition 9, we have P is a Gwsq-primary ideal of R and whenever $x,y\in h\left(R\right)$, $xy=0$, $x^2\notin P$ and $y\notin Grad\left(P\right)$, then $x,y\in An{n}_R\left(N\right)$. Conversely, let $(0,0)\ne (x,m)(y,s)\in P(+)N$, for some $(x,m),(y,s)\in h\left(R\right(+\left)M\right)$. Then $xy\in P$. If $xy\ne 0$, then we have $x^2\in P$ or $y\in Grad\left(P\right)$. If $x^2\in P$, then we have $x\in Grad\left(P\right)$, which implies that $2xm\in Grad\left(P\right)M\subseteq N$. Then ${(x,m)}^2=(x^2,2xm)\in P(+)N$. If $y\in Grad\left(P\right)$, then we have $(y,s)\in Grad\left(P\right)(+)M=Grad\left(P\right(+\left)N\right)$, this means that $P(+)N$ is a Gwsq-primary ideal of $R(+)M$. Now, assume that $xy=0$. If $x^2\notin P$ and $y\notin Grad\left(P\right)$, then $x,y\in An{n}_R\left(N\right)$. Since $xs+ym\in N$, we conclude that $x(xs+ym)=0$, which implies $x^{2}s=0$. As M is graded reduced, we have $xs=0$. Similarly, $ym=0$. Then we have $(x,m)(y,s)=(0,0)$, which is a contradiction. □

4. Conclusions

This article introduces and examines the concept of Gwsq-primary ideals. Assuming G is a group and R is a commutative G-graded ring with nonzero unity. A suitable graded ideal P of R is said to be a Gwsq-primary ideal if whenever $0\ne ab\in P$, for some $a,b\in h\left(R\right)$, then $a^2\in P$ or $b^n\in P$ for some positive integer n. Several examples and attributes of Gwsq-primary ideals are provided. We compared Gwsq-primary ideals with other classical graded ideals, such as graded strongly quasi primary ideals, graded weakly primary ideals, and graded weakly 2-prime ideals. As a proposal for future work, we will propose and study Gwsq-primary ideals over non-commutative graded rings. We propose the following definition: Let R be a non-commutative graded ring with nonzero unity and P be a proper graded ideal of R. Then P is said to be a Gwsq-primary ideal if whenever $0\ne xRy\subseteq P$, for some $x,y\in h\left(R\right)$, then $x^2\in P$ or $y^n\in P$, for some positive integer n. Note that, if R is commutative, then this definition coincides with Gwsq-primary ideals over commutative graded rings. However, we will try to prove that it is not the same case in non-commutative graded rings.