The ideal sheaves on a geometric object are closely connected to its subspaces.
Then, the support Z of OX/J is a closed subspace of X, and (Z, OX/J) is a scheme (both assertions can be checked locally).
Then, the kernel J of i# is a quasi-coherent ideal sheaf, and i induces an isomorphism from Z onto the closed subscheme defined by J.
[1] A particular case of this correspondence is the unique reduced subscheme Xred of X having the same underlying space, which is defined by the nilradical of OX (defined stalk-wise, or on open affine charts).
This ideal sheaf also gives A the structure of a reduced closed complex subspace.