dual of the duplication of a module along an ideal

Let r be a commutative ring with identity, i be an ideal of r, and m be an r-module. the amalgamated duplication of r along t, denoted by r bowtie i, is the special subring of r × r defined by r bowtie i := {(a, a + i) | a ∈ r, i ∈ i}. in this paper, we introduce and investigat a special kind of r bowtie i-modules, called the dual of the duplication of m along i  defined by mbowtied i= {(m1, m2) ∈ m × m | m1 - m2 × (0 :m annr(i))}. in fact the r bowtie i-module mbowtied i is a dual of the r bowtie i-module m bowtie i := {(m,m') ∈ m × m | m - m' ∈ im} which is introduced and studied by e. m. bouba, n. mahdou and m. tamekkante.