Write M = _{I C} I. We need to verify:

0 M.

If x, y M, then x + y M.

For each r R and m M, we have rm M.

Each I C is an ideal and so contains 0. Hence 0 _{I C} I = M.

Each I C is an ideal containing both x and y, whence x + y, and so x + y _{I C} I = M.

Each I C is an ideal containing m and hence rm. Therefore M = _{I C} I also contains rm.