Prime and weakly prime submodules on amalgamated duplication of a ring along an ideal

Let $A$ be a commutative ring with identity. A proper submodule $N$ of $A$-module $M$ is said to be prime submodule if $ax \in N$ where $a \in A, x \in M$, implies $x \in N$ or $aM \subseteq N$. A proper submodule $N \subset M$ is said to be weakly prime submodule if $0 \neq ax \in N$ where $a \in A, x \in M$, then either $x \in N$ or $aM \subseteq N$. The notion of weakly prime submodule was introduced by Atani and Farzalipour \cite{atani2007weakly}. The purpose of this paper is to study the form of prime and weakly prime submodules of duplication of the $A$-module $M$ along the ideal $I$ (denoted by $M \bowtie I$), introduced and studied by E. M. Bouba, N. Mahdou and M. Tamekkante. A number of results concerning prime and weakly prime submodules on amalgamated duplication and examples are given.