On graded weakly $J_{gr}$-semiprime submodules

1.   Introduction

Throughout this work, we assume that $\mathcal{A}$ is a commutative $\Gamma$-graded ring with identity and $\mathcal{D}$ is a unitary graded $\mathcal{A}$-module.

The study of graded rings and modules has attracted the attentions of many researchers for a long time due to their important applications in many fields in such as geometry and physics. For example, graded Lie algebra plays a significant role in differential geometry, such as with Frolicher-Nijenhuis, as well as the Nijenhuis-Richardson bracket (see ^[12]). In addition, they solve many physical problems related to supermanifolds, supersymmetries and quantizations of systems with symmetry (see ^[6,17]). Recently, some classical notions and definitions have been extended and generalized. For instance: the concepts of graded weakly semiprime ideals have been extended to the concepts of graded weakly semiprime submodules (see ^{[2,9,10,13,18]}). The main goal of this paper is to study the theory of graded modules over graded commutative rings. In particular, we introduce graded weakly $J_{gr}$ -semiprime submodules, which are a generalization of graded weakly semiprime submodules. Also, several results concerning graded weakly $J_{gr}$-semiprime submodules will be given.

Let $\Gamma$ be a group. A ring $\mathcal{A}$ is said to be a $\Gamma$-graded ring if there exist additive subgroups $\mathcal{A} _{g}$ of $\mathcal{A}$ indexed by the elements $g\in \Gamma$ with $\mathcal{A} = \bigoplus_{g\in \Gamma}\mathcal{A} _{g}$ and $\mathcal{A} _{g}\mathcal{A} _{h}\subseteq \mathcal{A} _{gh}$ for all $g$, $h\in \Gamma$. We set $h(\mathcal{A}): = \cup _{g\in \Gamma}\mathcal{A} _{g}$. If $t\in \mathcal{A}$, then $t$ can be written uniquely as $\sum_{g\in \Gamma}t_{g}$, where $t_{g}$ is called a homogeneous component of $t$ in $\mathcal{A} _{g}$. Let $\mathcal{A} = \bigoplus_{g\in \Gamma}\mathcal{A} _{g}$ be a $\Gamma$-graded ring. An ideal $L$ of $\mathcal{A}$ is said to be a graded ideal if $L = \bigoplus_{g\in \Gamma}(L\cap \mathcal{A} _{g}): = \bigoplus_{g\in \Gamma}L_{g}$. By $L\leq^{id} _{\Gamma}\mathcal{A}$, we mean that $L$ is a graded ideal of $\mathcal{A}$. Also, by $L < ^{id}_{\Gamma}\mathcal{A}$, we mean that $L$ is a proper graded ideal of $\mathcal{A}$. Let $\mathcal{A}$ be a $\Gamma$-graded ring, and $\mathcal{D}$ an $\mathcal{A}$-module. Then, $\mathcal{D}$ is a $\Gamma$-graded $\mathcal{A}$-module if there exists a family of additive subgroups $\{\mathcal{D} _{g}\}_{g\in \Gamma}$ of $\mathcal{D}$ with $\mathcal{D} = \bigoplus_{g\in \Gamma}\mathcal{D} _{g}$ and $\mathcal{A} _{g}\mathcal{D} _{h}\subseteq \mathcal{D} _{gh}$ for all $g, h\in \Gamma$. We set $h(\mathcal{D}): = \cup _{g\in \Gamma\ }\mathcal{D} _{g}.$ Let $\mathcal{D} = \bigoplus_{g\in \Gamma}\mathcal{D} _{g}$ be a graded $\mathcal{A}$-module. A submodule $U$ of $\mathcal{D}$ is said to be a graded submodule of$M$ if $U = \bigoplus_{g\in \Gamma}(U\cap \mathcal{D} _{g}): = \bigoplus_{g\in \Gamma}U_{g}$. By $U\leq^{sub} _{\Gamma}\mathcal{D}$, we mean that $U$ is a $\Gamma$-graded submodule of $\mathcal{D}$. Also, by $U < ^{sub}_{\Gamma}\mathcal{D}$, we mean that $U$ is a proper $\Gamma$-graded submodule of $\mathcal{D}.$ These basic properties and more information on graded rings and modules can be found in ^{[11,14,15,16]}. A $< ^{sub}_{\Gamma}\mathcal{D}$ is said to be a $Gr$-maximalif there is a $L\leq^{sub} _{\Gamma}\mathcal{D}$ with $U\subseteq L\subseteq \mathcal{D},$ and then $U = L$ or $L = \mathcal{D}$ (see ^[16]). The graded Jacobson radical of a graded module $\mathcal{D}$, denoted by $J_{gr}(\mathcal{D})$, is defined to be the intersection of all $Gr$-maximal submodules of $\mathcal{D}$ (if $\mathcal{D}$ has no $Gr$-maximal submodule then we shall take, by definition, $J_{gr}(\mathcal{D}) = \mathcal{D})$, (see ^[16]). A $U < ^{sub}_{\Gamma}\mathcal{D}$ is called a graded semiprime(briefly, $Gr$-semiprime) submoduleif, whenever $t_{g}\in h(\mathcal{A})$, $m_{h}\in h(\mathcal{D})$ and $n\in \mathbb{Z}^{+}$ with $t_{g}^{n}m_{h}\in U$, then $t_{g}m_{h}\in U$ (see ^[10]). A $U < ^{sub}_{\Gamma}\mathcal{D}$ is called a graded weakly semiprime (briefly, $Gr$-$W$-semiprime) submoduleif whenever $t_{g}\in h(\mathcal{A})$, $m_{h}\in h(\mathcal{D})$ and $n\in \mathbb{Z}^{+}$ with $0\neq \ t_{g}^{n}m_{h}\in U$, then $t_{g}m_{h}\in U$ (see ^[18]). It is shown in [4, Lemma 2.11] that if $U\leq^{sub} _{\Gamma}\mathcal{D}$, then $(U:_{\mathcal{A}}\mathcal{D}) = \{r\in \mathcal{A} :rU\subseteq \mathcal{D}\}$ is a graded ideal of $\mathcal{A}$. Let $N\leq^{sub} _{\Gamma}\mathcal{D}$ and $I\leq^{id} _{\Gamma}\mathcal{A}$. We use the notation $(N:_{\mathcal{D}}I)$ to denote the graded submodule $\{m\in \mathcal{D}:Im\subseteq N\}$ of $\mathcal{D}$.

2.   Results

Definition 2.1. A proper graded submodule $U$ of $\mathcal{D}$ is said to be a graded weakly $J_{gr}$-semiprime (briefly, $Gr$-$W$-$J_{gr}$-semiprime) submodule of $\mathcal{D}$ if, whenever $0\neq r_{g}^{n}m_{h}\in U$ where $r_{g}\in h(\mathcal{A})$, $m_{h}\in h(\mathcal{D})$ and $n\in \mathbb{Z} ^{+}$, then $r_{g}m_{h}\in U+J_{gr}(\mathcal{D})$. In particular, a graded ideal $L$ of $\mathcal{A}$ is said to be a graded weakly $J_{gr}$-semiprime ideal of $\mathcal{A}$ if $L$ is a graded weakly $J_{gr}$-semiprime submodule of the graded $\mathcal{A}$-module $\mathcal{A}$.

It is clear that every $Gr$-$W$-semiprime submodule is a $Gr$-$W$-$J_{gr}$-semiprime submodule of $\mathcal{D}$, but the converse is not true in general. This is clear from the following examples.

Example 2.2. Let $\Gamma = \mathbb{Z} _{2}$ and $\mathcal{A} = \mathbb{Z}$ be a $\Gamma$-graded ring with $\mathcal{A} _{0} = \mathbb{Z}$, $\mathcal{A} _{1} = \{0\}$. Then $\mathcal{D} = \mathbb{Z} _{24}$ is a graded $\mathcal{A}$-module with $\mathcal{D} _{0} = \mathbb{Z} _{24}$ and $\mathcal{D} _{1} = \{\overline{0}\}$. Let $U = \{\overline{0}, \overline{8}, \overline{16}\}$ $\leq^{sub} _{\Gamma}$ $\mathbb{Z} _{24}$. Since $J_{gr}(\mathbb{Z} _{24}) = \langle \overline{2}\rangle \cap \langle \overline{3}\rangle = \langle \overline{6}\rangle = \{\overline{0}, \overline{6}, \overline{12}, \overline{18} \}$, and whenever $0\neq r^{k}m\in U$ for $r\in h(\mathbb{Z})$, $m\in h(\mathbb{Z} _{24})$ and $k\in \mathbb{Z} ^{+}$ implies that $rm\in U+J_{gr}(\mathbb{Z} _{24}) = \{\overline{0}, \overline{8}, \overline{16}\}+\{\overline{0}, \overline{6 }, \overline{12}, \overline{18}\} = \langle \overline{2}\rangle,$ we have $U$ is a $Gr$-$W$-$J_{gr}$-semiprime submodule of $\mathcal{D}$. However, $U$ is not $a Gr$-$W$-semiprime submodule of $\mathcal{D}$ since there exist $2\in h(\mathbb{Z})$, $\overline{2}\in h(\mathbb{Z} _{24})$, and $2\in \mathbb{Z} ^{+}$ such that $0\neq 2^{2}\cdot \overline{2} = \overline{8}\in U$, but $2\cdot \overline{2} = \overline{4}\notin U$.

Example 2.3. Let $G = \mathbb{Z} _{2}$ and $R = \mathbb{Z}$ be a $G$-graded ring with $R_{0} = \mathbb{Z}$ and $R_{1} = \{0\}$. Let $M = \mathbb{Z} _{p^{_{\infty }}} = \{\frac{a}{p^{n}}+ \mathbb{Z} :a, n\in \mathbb{Z},$ $n\geqslant 0\}$ be a graded $R$-module with $M_{0} = \mathbb{Z} _{p^{_{\infty }}}$ and $M_{1} = \{0_{ \mathbb{Z} _{p^{_{\infty }}}}\} = \{ \mathbb{Z} \},$ where $p$ is a fixed prime number. Consider the graded submodule $N = \langle \frac{1}{p^{3}}+ \mathbb{Z} \rangle$ of $M$. Then $N$ is not a $Gr$-$W$-semiprime submodule of $M$, since $0\neq p^{l}(\frac{1}{p^{3+l}}+ \mathbb{Z}) = \frac{1}{p^{3}}+ \mathbb{Z} \in N$ but $p(\frac{1}{p^{3+l}}+ \mathbb{Z})\notin N$, where $1\neq l\in \mathbb{Z} ^{+}$. However, easy computations show that $N$ is a $Gr$-$W$-$J_{gr}$ -semiprime submodule of $M$.

Following are theorems that give some equivalent characterizations of the $Gr$-$W$-$J_{gr}$-semiprime submodule.

Theorem 2.4. Let $U < ^{sub}_{\Gamma}\mathcal{D}$. Then the following statements are equivalent.

(ⅰ) $U$ is a $Gr$-$W$-$J_{gr}$-semiprime submodule of $\mathcal{D}$.

(ⅱ) $(U:_{\mathcal{D}}\langle r_{g}^{n}\rangle)\subseteq (\langle 0\rangle :_{\mathcal{D}}\langle r_{g}^{n}\rangle)\cup (U+J_{gr}(\mathcal{D}):_{\mathcal{D}}\langle r_{g}\rangle)$, for each $r_{g}\in h(\mathcal{A})$.

(ⅲ) Either $(U:_{\mathcal{D}}\langle r_{g}^{n}\rangle)\subseteq (\langle 0\rangle :_{\mathcal{D}}\langle r_{g}^{n}\rangle)$ or $(U:_{\mathcal{D}}\langle r_{g}^{n}\rangle)\subseteq (U+J_{gr}(\mathcal{D}):_{\mathcal{D}}\langle r_{g}\rangle)$, for each $r_{g}\in h(\mathcal{A})$.

Proof. $(i)\rightarrow (ii)$: Let $r_{g}\in h(\mathcal{A})$ and $m_{h}\in (U:_{\mathcal{D} }\langle r_{g}^{n}\rangle)\cap h(\mathcal{D}).$ Then $\langle r_{g}^{n}\rangle m_{h}\subseteq U$, and hence $r_{g}^{n}m_{h}\in U$. If $r_{g}^{n}m_{h}\neq 0,$ then $r_{g}m_{h}\in U+J_{gr}(\mathcal{D})$ as $U$ is a $Gr$-$W$-$J_{gr}$-semiprime submodule of $\mathcal{D}$. Hence, $\langle r_{g}\rangle m_{h}\subseteq U+J_{gr}(\mathcal{D})$, and it follows that $m_{h}\in (U+J_{gr}(\mathcal{D}):_{\mathcal{D} }\langle r_{g}\rangle)$. Thus, $m_{h}\in (\langle 0\rangle :_{\mathcal{D} }\langle r_{g}^{n}\rangle)\cup (U+J_{gr}(\mathcal{D}):_{\mathcal{D} }\langle r_{g}\rangle)$. If $r_{g}^{n}m_{h} = 0,$ then $\langle r_{g}^{n}\rangle m_{h}\subseteq \{0\}$, and so $m_{h}\in (\langle 0\rangle :_{\mathcal{D} }\langle r_{g}^{n}\rangle)$. Hence, $m_{h}\in (\langle 0\rangle :_{\mathcal{D} }\langle r_{g}^{n}\rangle)\cup (U+J_{gr}(\mathcal{D}):_{\mathcal{D} }\langle r_{g}\rangle)$. Therefore, $(U:_{\mathcal{D} }\langle r_{g}^{n}\rangle)\subseteq (\langle 0\rangle :_{\mathcal{D} }\langle r_{g}^{n}\rangle)\cup (U+J_{gr}(\mathcal{D}):_{\mathcal{D} }\langle r_{g}\rangle)$.

$(ii)\rightarrow (iii)$: It is clear.

$(iii)\rightarrow (i)$: Let $r_{g}\in h(\mathcal{A})$, $m_{h}\in h(\mathcal{D})$, and $n\in \mathbb{Z} ^{+}$ with $0\neq r_{g}^{n}m_{h}\in U$. Then $\{0\}\neq \langle r_{g}^{n}\rangle m_{h}\subseteq U$, which implies that $m_{h}\in (U:\langle r_{g}^{n}\rangle)$ and $m_{h}\notin (\langle 0\rangle :_{\mathcal{D} }\langle r_{g}^{n}\rangle).$ Now, by (iii), we get $m_{h}\in (U+J_{gr}(\mathcal{D}):_{\mathcal{D} }\langle r_{g}\rangle)$ and so $r_{g}m_{h}\in U+J_{gr}(\mathcal{D})$. Therefore, $U$ is a $Gr$-$W$-$J_{gr}$-semiprime submodule of $\mathcal{D}$. □

Theorem 2.5. Let $U < ^{sub}_{\Gamma}\mathcal{D}$. Then the following statements are equivalent.

(ⅰ) $U$ is a $Gr$-$W$-$J_{gr}$-semiprime submodule of $\mathcal{D}$.

(ⅱ) For every $K\leq^{sub} _{\Gamma}\mathcal{D},$ $r_{g}\in h(\mathcal{A})$, and $n\in \mathbb{Z} ^{+}$ with $\{0\}\neq \langle r_{g}\rangle ^{n}K\subseteq U,$ then $\langle r_{g}\rangle K\subseteq U+J_{gr}(\mathcal{D})$.

Proof.

$(i)\Rightarrow (ii)$ Let $K$ $\leq^{sub} _{\Gamma}\mathcal{D},$ $r_{g}\in h(\mathcal{A})$, and $n\in \mathbb{Z} ^{+}$ with $\{0\}\neq \langle r_{g}\rangle ^{n}K\subseteq U.$ This implies that, $K\subseteq (U:_{\mathcal{D}}\langle r_{g}^{n}\rangle)$ and $K\nsubseteq (\langle 0\rangle :_{\mathcal{D}}\langle r_{g}^{n}\rangle).$ Since $U$ is a $Gr$-$W$ -$J_{gr}$-semiprime submodule of $\mathcal{D}$, by Theorem 2.4 we have $K\subseteq$ $(U:_{\mathcal{D}}\langle r_{g}^{n}\rangle)\subseteq (U+J_{gr}(\mathcal{D}):_{\mathcal{D}}\langle r_{g}\rangle),$ hence $\langle r_{g}\rangle K\subseteq U+J_{gr}(\mathcal{D})$.

$(ii)\Rightarrow (i)$ Let $0\neq r_{g}^{n}m_{h}\in U$ where $r_{g}\in h(\mathcal{A})$, $m_{h}\in h(\mathcal{D})$, and $n\in \mathbb{Z} ^{+}.$ Then $\{0\}\neq \langle r_{g}\rangle ^{n}\langle m_{h}\rangle \subseteq U$. Now, by (ii), we have $\langle r_{g}\rangle \langle m_{h}\rangle \subseteq U+J_{gr}(\mathcal{D}),$ and it follows that $r_{g}m_{h}\in U+J_{gr}(\mathcal{D}).$ Therefore, $U$ is a $Gr$-$W$-$J_{gr}$-semiprime submodule of $\mathcal{D}.$ □

The following corollaries follow directly from Theorem 2.5.

Corollary 2.6. Let $U < ^{sub}_{\Gamma}\mathcal{D}.$ Then $U$ is a $Gr$ -$W$-$J_{gr}$-semiprime submodule of $\mathcal{D}$ if and only if for every $r_{g}\in h(\mathcal{A})$, and $n\in \mathbb{Z} ^{+}$ with $\{0\}\neq \langle r_{g}^{n}\rangle \mathcal{D}\subseteq U$, then $\langle r_{g}\rangle \mathcal{D}\subseteq U+J_{gr}(\mathcal{D})$.

Corollary 2.7. Let $U < ^{sub}_{\Gamma}\mathcal{D}.$ Then $U$ is a $Gr$-$W$-$J_{gr}$-semiprime submodule of $\mathcal{D}$ if and only if for every $r_{g}\in h(\mathcal{A})$, $K\leq^{sub} _{\Gamma}\mathcal{D}$ and $n\in \mathbb{Z} ^{+}$ with $\{0\}\neq r_{g}^{n}K\subseteq U$, then $r_{g}K\subseteq U+J_{gr}(\mathcal{D})$.

Theorem 2.8. Let $U$ be a $Gr$-$W$-$J_{gr}$ -semiprime submodule of $\mathcal{D}$ with $J_{gr}(\mathcal{D})\subseteq U.$ Then $U$ is a $Gr$-$W$-semiprime submodule of $\mathcal{D}$.

Proof. Let $r_{g}\in h(\mathcal{A})$, $m_{h}\in h(\mathcal{D})$, and $n\in \mathbb{Z} ^{+}$ with $0\neq r_{g}^{n}m_{h}\in U.$ Since $U$ is a $Gr$-$W$-$J_{gr}$ -semiprime submodule of $\mathcal{D}$ and $J_{gr}(\mathcal{D})\subseteq U$, we have $r_{g}m_{h}\in U+J_{gr}(\mathcal{D}) = U$. Therefore, $U$ is a $Gr$-$W$-semiprime submodule of $\mathcal{D}$. □

Let $\mathcal{A}$ be a $\Gamma$-graded ring and $\mathcal{D}$, $\mathcal{D} ^{^{\prime }}$ be two graded $\mathcal{A}$-modules. Let $\varphi :\mathcal{D} \rightarrow \mathcal{D} ^{^{\prime }}$ be an $\mathcal{A}$-module homomorphsim. Then, $\varphi$ is said to be a graded homomorphsim if $\varphi (\mathcal{D} _{g})\subseteq \mathcal{D} _{g}^{^{\prime }}$ for all $g\in \Gamma$, see ^[16].

Theorem 2.9. Let $U < _{\Gamma }^{sub}\mathcal{D}$. If $J_{gr}(\mathcal{D} /U) = \{U\}$, then $J_{gr}(\mathcal{D})\subseteq U$.

Proof. Define $\varphi :\mathcal{D} \rightarrow \mathcal{D} /U$ as a graded homeomorphism given by $\varphi (x) = x+U$ for all $x\in h(\mathcal{D})$; by [3, Theorem 2.12 (ⅰ)], $\varphi (J_{gr}(\mathcal{D}))\subseteq J_{gr}(\mathcal{D} /U)$. Since $J_{gr}(\mathcal{D} /U) = \{U\}$, then $\{U\}\subseteq \varphi (J_{gr}(\mathcal{D}))\subseteq \{U\}$, so we have $\varphi (J_{gr}(\mathcal{D})) = \{U\}$, thus $J_{gr}(\mathcal{D})\subseteq Ker\varphi = U$. □

Corollary 2.10. Let $U$ be a $Gr$-$W$-$J_{gr}$-semiprime submodule of $\mathcal{D}$ with $J_{gr}(\frac{\mathcal{D}}{U}) = \{U\}$. Then $U$ is a $Gr$-$W$-semiprime submodule of $\mathcal{D}$.

Proof. This is clear by Theorems 2.9 and 2.8. □

A graded $\mathcal{A}$-module $\mathcal{D}$ is a graded semisimple ($Gr$-semisimple) if and only if every graded submodule $U$ of $\mathcal{D}$ is a direct summand. That is $\mathcal{D}$ is a $Gr$-semisimple if and only if for every graded submodule $U$ of $\mathcal{D}$ there exists $L$, a graded submodule of $\mathcal{D}$ such that $\mathcal{D} = U\oplus L$.

A graded submodule $U$ is called a graded small ($Gr$-small) if $\mathcal{D} = U+V$ for $V$ $\leq^{sub} _{\Gamma}$ $\mathcal{D}$ implies that $V = \mathcal{D}$, see ^[1].

Theorem 2.11. Let $\mathcal{D}$ be a $Gr$-semisimple $\mathcal{A}$-module and $U$ be a $Gr$-$W$-$J_{gr}$-semiprime submodule of $\mathcal{D}$. Then $U$ is a $Gr$-$W$-semiprime submodule of $\mathcal{D}$.

Proof. Let $\mathcal{D}$ be a $Gr$-semisimple $\mathcal{A}$-module. Then every graded submodule of $\mathcal{D}$ is a direct summand. Thus, the only $Gr$-small sumodule of $\mathcal{D}$ is $\{0\}$, and it follows that $J_{gr}(\mathcal{D}) = \sum \{S:S$ is a $Gr$-small submodule of $\mathcal{D} \} = \{0\}\subseteq U$ by [3, Theorem 2.10]. Since $U$ is a $Gr$-$W$-$J_{gr}$-semiprime submodule of $\mathcal{D}$, by Theorem 2.8, then $U$ is a $Gr$-$W$-semiprime submodule of $\mathcal{D}$. □

Recall from ^[4] that a graded module $\mathcal{D}$ is said to be a graded torsion ($Gr$-torsion) free $\mathcal{A}$-module if, whenever $r_{g}m_{h} = 0$ where $r_{g}\in h(\mathcal{A})$ and $m_{h}\in$h($\mathcal{D}$), then either $m_{h} = 0$ or $r_{g} = 0$.

Theorem 2.12. Let $\mathcal{D}$ be a $Gr$-torsion free $\mathcal{A}$ -module, and $U\leq^{sub} _{\Gamma}\mathcal{D}$ with $J_{gr}(\frac{\mathcal{D}}{U}) = \{U\}$. Then $U$ is a $Gr$-$W$-$J_{gr}$-semiprime submodule of $\mathcal{D}$ if and only if for any nonzero $L$ $\leq^{id} _{\Gamma}$ $\mathcal{A}$, $(U:_{\mathcal{D}}L)$ is a $Gr$-$W$-$J_{gr}$ -semiprime submodule of $\mathcal{D}$.

Proof. $(\Longrightarrow)$ Let $0\neq L$ $\leq^{id} _{\Gamma}$ $\mathcal{A}$, $m_{h}\in h(\mathcal{D})$, $r_{g}\in h(\mathcal{A})$, and $n\in \mathbb{Z} ^{+}$ with $0\neq r_{g}^{n}m_{h}\in (U:_{\mathcal{D} }L).$ Then $\{0\}\neq \langle r_{g}^{n}\rangle m_{h}\subseteq (U:_{\mathcal{D} }L),$ and hence $\langle r_{g}^{n}\rangle (Lm_{h})\subseteq U$. If $\langle r_{g}^{n}\rangle (Lm_{h}) = \{0\},$ so there exists $0\neq i\in L\cap h(\mathcal{A})$ with $\langle r_{g}^{n}\rangle im_{h} = \{0\},$ so $i\cdot r_{g}^{n}m_{h} = 0$. Hence, $r_{g}^{n}m_{h} = 0$ as $\mathcal{D}$ is a $Gr$-torsion free $\mathcal{A}$-module, which is a contradiction. So, assume that $\langle r_{g}\rangle ^{n}(Lm_{h}) = \langle r_{g}^{n}\rangle (Lm_{h})\neq \{0\}.$ Since $U$ is a $Gr$-$W$-$J_{gr}$ -semiprime submodule of $\mathcal{D},$ by Theorem 2.5, $\langle r_{g}\rangle (Lm_{h})\subseteq U+J_{gr}(\mathcal{D}).$ But, $J_{gr}(\frac{\mathcal{D} }{U}) = \{U\}$, and by Theorem 2.9, we have $J_{gr}(\mathcal{D})\subseteq U$ so $\langle r_{g}\rangle (Lm_{h})\subseteq U.$ This implies that, $\langle r_{g}\rangle m_{h}\subseteq (U:_{\mathcal{D} }L)\subseteq (U:_{\mathcal{D} }L)+J_{gr}(\mathcal{D})$ and hence $r_{g}m_{h}\in (U:_{\mathcal{D} }L)\subseteq (U:_{\mathcal{D} }L)+J_{gr}(\mathcal{D}, ).$ Therefore, $(U:_{\mathcal{D} }L)$ is a $Gr$-$W$-$J_{gr}$-semiprime submodule of $\mathcal{D}$.

$(\Longleftarrow)$ Assume that $(U:_{\mathcal{D} }L)$ is a $Gr$-$W$-$J_{gr}$ -semiprime submodule of $\mathcal{D}$ for any nonzero $L$ $\leq^{id} _{\Gamma}$ $\mathcal{A}$. Put $\mathcal{A} = L,$ then $U =$ $(U:_{\mathcal{D} }\mathcal{A})$ is a $Gr$-$W$-$J_{gr}$-semiprime submodule of $\mathcal{D}.$ □

Recall from ^[7] that a graded $\mathcal{A}$-module $\mathcal{D}$ is called a graded multiplication module ($Gr$-multiplication module) if for every $U$ $\leq^{sub} _{\Gamma}$ $\mathcal{D}$ there exists a $K$ $\leq^{id} _{\Gamma}$ $\mathcal{A}$ such that $U = K\mathcal{D}.$ If $\mathcal{D}$ is a $Gr$-multiplication $\mathcal{A}$-module, $U = (U:_{\mathcal{A} }\mathcal{D})\mathcal{D}$ for every $U$ $\leq^{sub} _{\Gamma}$ $\mathcal{D}$.

The set of all homogeneous zero divisors of $\mathcal{A}$ is $G$-$Z(\mathcal{A}) = \{r\in h(\mathcal{A}):$ $rs = 0$ for some $0\neq s\in h(\mathcal{A})\}$, and the set of all homogeneous regular elements is $G$-$C(\mathcal{A}) = \{c\in h(\mathcal{A}):c\notin G$-$Z(\mathcal{A})\} = \{c\in h(\mathcal{A}):cr\neq 0$ for all $0\neq r\in h(\mathcal{A})\}$. It is clear that $\mathcal{D}$ is a $Gr$-torsion free if and only if $cm\neq 0$ for all $c\in G$-$C(\mathcal{A})$ and $0\neq m\in h(\mathcal{D})$.

Theorem 2.13. Every faithful $Gr$-multiplication $\mathcal{A}$-module is a $Gr$-torsion free.

Proof. Suppose that, $\mathcal{D}$ is not $Gr$-torsion free. Hence, there exist $c\in G$-$C(\mathcal{A})$ and $0\neq m\in h(\mathcal{D})$ with $cm = 0.$ Since $\mathcal{D}$ is a faithful $Gr$-multiplication $\mathcal{A}$-module, there exists an $L$ $\leq^{id} _{\Gamma}$ $\mathcal{A}$ with $\mathcal{A} m = L\mathcal{D},$ and so $\mathcal{A} cm = cL\mathcal{D}.$ This implies that $(cL)\mathcal{D} = \{0\}$. Since $\mathcal{D}$ is a faithful, $cL = \{0\}$. Hence, $c\in G$-$Z(\mathcal{A})$ since $L\neq0$, and so $c\notin G$-$C(\mathcal{A})$, which is a contradiction. Therefore, $\mathcal{D}$ is $Gr$ -torsion free.□

Corollary 2.14. Let $\mathcal{D}$ be a faithful $Gr$ -multiplication $\mathcal{A}$-module and $U\leq^{sub} _{\Gamma}\mathcal{D}$, with $J_{gr}(\frac{\mathcal{D}}{U}) = \{U\}$. Then, $U$ is a $Gr$-$W$-$J_{gr}$-semiprime submodule of $\mathcal{D}$ if and only if for any nonzero $L$ $\leq^{id} _{\Gamma}$ $\mathcal{A}$, $(U:_{\mathcal{D}}L)$ is a $Gr$ -$W$-$J_{gr}$-semiprime submodule of $\mathcal{D}$.

Proof. Follows by Theorems 2.13 and 2.12. □

Theorem 2.15. Let $U$ be $Gr$-small submodule of $\mathcal{D}$ with $J_{gr}(\mathcal{D})$ a $Gr$-$W$-semiprime submodule of $\mathcal{D}$. Then, $U$ is a $Gr$-$W$-$J_{gr}$-semiprime submodule of $\mathcal{D}$.

Proof. Let $r_{g}\in h(\mathcal{A}),$ $m_{h}\in h(\mathcal{D})$, and $n\in \mathbb{Z} ^{+}$ with $0\neq r_{g}^{n}m_{h}\in U$. Since $U$ is a $Gr$-small submodule of $\mathcal{D}$, then by [3,Theorem 2.10], $U\subseteq J_{gr}(\mathcal{D}) = \sum \{A$ $:$ $A$ is a $Gr$-small submodule of $\mathcal{D}\}$, so $0\neq r_{g}^{n}m_{h}\in J_{gr}(\mathcal{D})$, since $J_{gr}(\mathcal{D})$ is a $Gr$-$W$-semiprime submodule of $\mathcal{D}$, then $r_{g}m_{h}\in J_{gr}(\mathcal{D})\subseteq U+J_{gr}(\mathcal{D})$. Therefore $U$ is a $Gr$-$W$-$J_{gr}$ -semiprime submodule of $\mathcal{D}$. □

Recall from ^[16] that graded $\mathcal{A}$-module $\mathcal{D}$ is said to be a graded finitely generated ($Gr$-finitely generated) if $\mathcal{D} = \mathcal{A} a_{g1}+ \cdot +\mathcal{A} a_{gn}$ for some $a_{g1}, a_{g2}, ..., a_{gn}\in h(\mathcal{D}).$

Theorem 2.16. Let $\mathcal{D}$ be a $Gr$-finitely generated $Gr$-multiplication $\mathcal{A}$-module and $L$ be a $Gr$-$W$-$J_{gr}$-semiprime ideal of $\mathcal{A}$ with $ann_{\mathcal{A}}(\mathcal{D})\subseteq L$. Then $L\mathcal{D}$ is a $Gr$-$W$-$J_{gr}$-semiprime submodule of $\mathcal{D}$.

Proof. Let $r_{g}\in h(\mathcal{A})$, $m_{h}\in h(\mathcal{D})$, and $k\in \mathbb{Z} ^{+}$ with $0\neq r_{g}^{k}m_{h}\in L\mathcal{D}$. Then $\{0\}\neq r_{g}^{k}\langle m_{h}\rangle \subseteq L\mathcal{D}.$ Since $\mathcal{D}$ is a $Gr$-multiplication, $\langle m_{h}\rangle = J\mathcal{D}$ for some $J\leq^{id} _{\Gamma}\mathcal{A}$, and hence $\{0\}\neq r_{g}^{k}J\mathcal{D} \subseteq L\mathcal{D}.$ This implies that $\{0\}\neq r_{g}^{k}J\subseteq L+ann_{\mathcal{A}}(\mathcal{D})$ by [5,Lemma 3.9]. Since $ann_{\mathcal{A}}(\mathcal{D})\subseteq L$, and it follows that $\{0\}\neq r_{g}^{k}J\subseteq L$. Hence, $r_{g}J\subseteq L+J_{gr}(\mathcal{A})$ as $L$ is a $Gr$-$W$-$J_{gr}$-semiprime ideal of $\mathcal{A}.$ Thus, $r_{g}J\mathcal{D} \subseteq L\mathcal{D} +J_{gr}(\mathcal{A})\mathcal{D} \subseteq L\mathcal{D} +J_{gr}(\mathcal{D})$, and so $r_{g}m_{h}\in r_{g}\langle m_{h}\rangle \subseteq L\mathcal{D} +J_{gr}(\mathcal{D})$. Therefore, $L\mathcal{D}$ is a $Gr$-$W$-$J_{gr}$-semiprime submodule of $\mathcal{D}$. □

The following example shows that the residual of the $Gr$-$W$-$J_{gr}$-semiprime submodule is not necessarily a $Gr$-$W$-$J_{gr}$-semiprime ideal.

Example 2.17. Let $\Gamma = \mathbb{Z} _{2}$ and $\mathcal{A} = \mathbb{Z}$ be a $\Gamma$-graded ring such that $\mathcal{A}_{0} = \mathbb{Z}$ and $\mathcal{A}_{1} = \{0\}$. Let $\mathcal{D} = \mathbb{Z} _{8}$ be a graded $\mathcal{A}$-module such that $\mathcal{D}_{0} = \mathbb{Z} _{8}$ and $\mathcal{D}_{1} = \{\overline{0}\}$. Let $U = \{\overline{0}, \overline{4} \} = \langle \overline{4}\rangle$ $\leq^{sub} _{\Gamma}\mathcal{D}$. Thus, it is a $Gr$-$W$- $J_{gr}$-semiprime submodule of $\mathcal{D}$ where $J_{gr}(\mathcal{D}) = \langle \overline{2}\rangle$. However $(U:_{\mathcal{A}}\mathcal{D}) = 4 \mathbb{Z}$ is not a $Gr$-$W$-$J_{gr}$-semiprime ideal of $\mathcal{A},$ since $0\neq 2^{2}\cdot 1\in (U:_{\mathcal{A}}\mathcal{D})$ where $2$, $1\in h(\mathcal{A})$, but $2\cdot 1\notin (U:_{\mathcal{A}}\mathcal{D})+J_{gr}(\mathcal{A}) = 4 \mathbb{Z} +(0) = 4 \mathbb{Z}.$

The following theorems show that the residual of a $Gr$-$W$-$J_{gr}$-semiprime submodule is a $Gr$-$W$-$J_{gr}$-semiprime ideal with under conditions.

Theorem 2.18. Let $\mathcal{D}$ be a $Gr$-faithful $\mathcal{A}$-module, and $U\leq^{sub} _{\Gamma}\mathcal{D}$ with $J_{gr}(\mathcal{D}/U) = \{U\}$ and $J_{gr}(\mathcal{A})\subseteq (U:_{\mathcal{A}}\mathcal{D})$. Then $U$ is a $Gr$-$W$-$J_{gr}$-semiprime submodule of $\mathcal{D}$ if and only if $(U:_{\mathcal{A}}\mathcal{D})$ is a $Gr$-$W$-$J_{gr}$-semiprime ideal of $\mathcal{A}$.

Proof. $(\Rightarrow)$ Let $a_{g}$, $b_{h}\in h(\mathcal{A})$ and $k\in \mathbb{Z} ^{+}$ with $0\neq a_{g}^{k}b_{h}\in (U:_{\mathcal{A}}\mathcal{D})$. Hence, $\{0\}\neq a_{g}^{k}b_{h}\mathcal{D}\subseteq U$. Since $U$ is a $Gr$-$W$ -$J_{gr}$-semiprime submodule of $\mathcal{D}$, by Corollary 2.7, we have $a_{g}b_{h}\mathcal{D}\subseteq U+J_{gr}(\mathcal{D}).$ By Theorem 2.9, we have $J_{gr}(\mathcal{D})\subseteq U$ since $J_{gr}(\mathcal{D}/U) = \{U\}.$ This implies that $a_{g}b_{h}\mathcal{D}\subseteq U.$ Thus, $a_{g}b_{h}\in (U:_{\mathcal{A}}\mathcal{D})\subseteq (U:_{\mathcal{A}}\mathcal{D})+J_{gr}(\mathcal{A})$. Therefore, $(U:_{\mathcal{A}}\mathcal{D})$ is a $Gr$-$W$-$J_{gr}$ -semiprime ideal of $\mathcal{A}$.

$(\Leftarrow)$ Let $r_{g}\in h(\mathcal{A})$ and $n\in \mathbb{Z} ^{+}$ with $\{0\}\neq \langle r_{g}\rangle ^{n}\mathcal{D}\subseteq U$. Hence, $\{0\}\neq \langle r_{g}\rangle ^{n}\subseteq (U:_{\mathcal{A}}\mathcal{D})$ (if $\langle r_{g}\rangle ^{n} = \{0\}$, then $\langle r_{g}\rangle ^{n}\mathcal{D} = \{0\}$ as a contradiction), it follows that $0\neq r_{g}^{n}.1\in (U:_{\mathcal{A}}\mathcal{D}).$ Hence. $r_{g}\cdot 1\in (U:_{\mathcal{A}}\mathcal{D})+J_{gr}(\mathcal{A})$ as $(U:_{\mathcal{A}}\mathcal{D})$ is a $Gr$-$W$-$J_{gr}$-semiprime ideal of $\mathcal{A}.$ Since $J_{gr}(\mathcal{A})\subseteq (U:_{\mathcal{A}}\mathcal{D})$, we have $r_{g}\in (U:_{\mathcal{A}}\mathcal{D})$, and it follows that $\langle r_{g}\rangle \subseteq (U:_{\mathcal{A}}\mathcal{D})$. This yields that $\langle r_{g}\rangle \mathcal{D}\subseteq U\subseteq U+J_{gr}(\mathcal{D})$. Thus < $U$ is a $Gr$-$W$ $J_{gr}$-semiprime submodule of $\mathcal{D}$ by Corollary 2.6. □

Recall that a graded $\mathcal{A}$-module $\mathcal{D}$ is said to be a graded cancellation ($Gr$-cancellation) if for any graded ideals $K$ and $L$ of $\mathcal{A}$, $K\mathcal{D} = L\mathcal{D}$, we have $K = L$, see ^[8].

Theorem 2.19. Let $\mathcal{D}$ be a $Gr$-finitely generated faithful $Gr$-multiplication $\mathcal{A}$-module and $U$ $< ^{sub}_{\Gamma}\mathcal{D}$. Then $U$ is a $Gr$-$W$-$J_{gr}$-semiprime submodule of $\mathcal{D}$ if and only if $(U:_{\mathcal{A}}\mathcal{D})$ is a $Gr$-$W$-$J_{gr}$-semiprime ideal of $\mathcal{A}$.

Proof. $(\Rightarrow)$ Let $a_{g}$, $b_{h}\in h(\mathcal{A})$ and $k\in \mathbb{Z} ^{+}$ with $0\neq a_{g}^{k}b_{h}\in (U:_{\mathcal{A}}\mathcal{D})$. Hence, $\{0\}\neq a_{g}^{k}b_{h}\mathcal{D}\subseteq U$. (if $a_{g}^{k}b_{h}\mathcal{D} = \{0\}$, then $a_{g}^{k}b_{h} = 0$ since $\mathcal{D}$ is a faithful as a contradiction). Since $U$ is a $Gr$-$W$-$J_{gr}$-semiprime submodule of $\mathcal{D}$, by Corollary 2.7 we get $a_{g}b_{h}\mathcal{D}\subseteq U+J_{gr}(\mathcal{D}).$ This implies that $a_{g}b_{h}\mathcal{D}\subseteq (U:_{\mathcal{A}}\mathcal{D})\mathcal{D}+J_{gr}(\mathcal{D})$ as $\mathcal{D}$ is a $Gr$-multiplication module. Since $J_{gr}(\mathcal{D}) = J_{gr}(\mathcal{A})\mathcal{D}$, we have $a_{g}b_{h}\mathcal{D}\subseteq (U:_{\mathcal{A}}\mathcal{D})\mathcal{D}+J_{gr}(\mathcal{A})\mathcal{D} = ((U:_{\mathcal{A}}\mathcal{D})+J_{gr}(\mathcal{A}))\mathcal{D}$, so $\langle a_{g}b_{h}\rangle \mathcal{D}\subseteq ((U:_{\mathcal{A}}\mathcal{D})+J_{gr}(\mathcal{A}))\mathcal{D}.$ Since $\mathcal{D}$ is a $Gr$-finitely generated faithful $Gr$-multiplication by [8,Theorem 2.10], we get $\langle a_{g}b_{h}\rangle \subseteq (U:_{\mathcal{A}}\mathcal{D})+J_{gr}(\mathcal{A})$. Hence, $\langle a_{g}b_{h}\rangle \subseteq (U:_{\mathcal{A}}\mathcal{D})+J_{gr}(\mathcal{A})$, so $a_{g}b_{h}\in (U:_{\mathcal{A}}\mathcal{D})+J_{gr}(\mathcal{A})$. Therefore, $(U:_{\mathcal{A}}\mathcal{D})$ is a $Gr$-$W$-$J_{gr}$-semiprime ideal of $\mathcal{A}$.

$(\Leftarrow)$ Let $\{0\}\neq r_{g}^{n}K\subseteq U$ where $r_{g}\in h(\mathcal{A})$ and $K\leq^{sub} _{\Gamma}\mathcal{D}$. Since $\mathcal{D}$ is a $Gr$-multiplication $\mathcal{A}$-module, then there exists nonzero $L\leq^{id} _{\Gamma}\mathcal{A}$ with $K = L\mathcal{D}$, and it follows that $\{0\}\neq r_{g}^{n}L\mathcal{D} \subseteq U$, hence $\{0\}\neq r_{g}^{n}L\subseteq (U:_{\mathcal{A}}\mathcal{D})$. So, $r_{g}L\subseteq (U:_{\mathcal{A}}\mathcal{D})+J_{gr}(\mathcal{A})$ as $(U:_{\mathcal{A}}\mathcal{D})$ is a $Gr$-$W$-$J_{gr}$-semiprime ideal of $\mathcal{A}.$ Hence, $r_{g}L\mathcal{D} \subseteq (U:_{\mathcal{A}}\mathcal{D})\mathcal{D} +J_{gr}(\mathcal{A})\mathcal{D} \subseteq (U:_{\mathcal{A}}\mathcal{D})\mathcal{D} +J_{gr}(\mathcal{D}).$ This implies that, $r_{g}K\subseteq U+J_{gr}(\mathcal{D})$ as $\mathcal{D}$ is a $Gr$-multiplication $\mathcal{A}$ -module. Thus, $U$ is a $Gr$-$W$-$J_{gr}$-semiprime submodule of $\mathcal{D}$ by Corollary 2.7. □

Theorem 2.20. Let $\mathcal{D}$ be a $Gr$-finitely generated faithful $Gr$-multiplication $\mathcal{A}$-module and $U$ $< ^{sub}_{\Gamma}\mathcal{D}$. Then the following statements are equivalent:

(ⅰ) $U$ is a $Gr$-$W$-$J_{gr}$-semiprime submodule of $\mathcal{D}$.

(ⅱ) $(U:_{\mathcal{A}}\mathcal{D})$ is a $Gr$-$W$-$J_{gr}$-semiprime ideal of $\mathcal{A}$.

(ⅲ) $U = L\mathcal{D}$ for some a $Gr$-$W$-$J_{gr}$-semiprime ideal $L$ of $\mathcal{A}$.

(i)$\Rightarrow$(ii). By Theorem 2.19,

[(ⅱ)$\Rightarrow$(ⅲ)] Since $\mathcal{D}$ is a $Gr$ -multiplication $\mathcal{A}$-module, $U = (U:_{\mathcal{A}}\mathcal{D})\mathcal{D}$, where $(U:_{\mathcal{A}}\mathcal{D})$ is a $Gr$-$W$- $J_{gr}$-semiprime ideal of $\mathcal{A}$.

[(ⅲ)$\Rightarrow$(ⅰ)] Let $U = L\mathcal{D}$ for some $Gr$-$W$-$J_{gr}$-semiprime ideal $L$ of $\mathcal{A}$. Let $\{0\}\neq \langle r_{g}\rangle ^{n}\mathcal{D} \subseteq U$ where $r_{g}\in h(\mathcal{A})$ and $n\in \mathbb{Z} ^{+}$, then $\{0\}\neq \langle r_{g}\rangle ^{n}\mathcal{D} \subseteq L\mathcal{D}.$ Since $\mathcal{D}$ is a $Gr$-finitely generated faithful $Gr$-multiplication, by [8, Theorem 2.10], $\mathcal{D}$ is a $Gr$-cancellation. Thus, $\{0\}\neq \langle r_{g}\rangle ^{n}\subseteq L$. Since $L$ is a $Gr$-$W$-$J_{gr}$-semiprime ideal of $\mathcal{A}$, $\langle r_{g}\rangle \subseteq L+J_{gr}(\mathcal{A})$, and it follows that $\langle r_{g}\rangle \mathcal{D} \subseteq L\mathcal{D} +J_{gr}(\mathcal{A})\mathcal{D}$. This yields that $\langle r_{g}\rangle \mathcal{D} \subseteq U+J_{gr}(\mathcal{D})$ since $J_{gr}(\mathcal{D}) = J_{gr}(\mathcal{A})\mathcal{D}$. Therefore, $U$ is a $Gr$-$W$-$J_{gr}$-semiprime submodule of $\mathcal{D}$. □

3.   Conclusions

In this paper, we introduced the concept of graded weakly $J_{gr}$-semiprime submodules of a graded module over a commutative graded ring, which is a generalization of graded weakly semiprime submodules. Also, we proved several properties as well as characterizations of graded weakly $J_{gr}$-semiprime submodules. Finally, we established the necessary and sufficient condition for graded submodules to be graded weakly $J_{gr}$-semiprime submodules.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors wish to thank sincerely the referees for their valuable comments and suggestions.

Conflict of interest

The authors declare no conflict of interest.