Supplement -coessential and supplement –coclosed submodules

Consider M as a left R−module where R is an associative ring with unity. We introduce a new classes of submodules called supplement -coessential and supplement –coclosed, as M is R −module and H, L are submodules of M such that 𝐻≤L≤M, then H is named supplement-coessential (in brief sp-coessential) submodule denoted by (H ≤sp.ce L in M in M) if LH≪spMH⁠, and M is R − module where L a submodule of M. We say that L is named supplement-coclosed submodule (in brief sp-coclosed) of M denoted by (L ≤sp.cc M) if whenever H ≤sp.ce L (i.e LH≤sp.ccMH⁠). for some submodule H of L, implies that H = L, respectively, also we study several properties related with these submodules.