Let K be an imaginary quadratic field where a prime number \(p \ge 5\) is inert. Let E be an elliptic curve defined over K and suppose that E has good supersingular reduction at p. In this paper, we prove that the plus/minus Selmer group of E over the anticyclotomic \({{\,\mathrm{\mathbb {Z}}\,}}_{p}\) -extension of K has no proper \(\Lambda \) -submodules of finite index under mild assumptions for E. This is an analogous result to R. Greenberg and B. D. Kim for the anticyclotomic \({{\,\mathrm{\mathbb {Z}}\,}}_{p}\) -extension essentially. By applying the results of A. Agboola–B. Howard or A. Burungale–K. Büyükboduk–A. Lei, we can also construct examples satisfying the assumptions of our theorem.
