 A projective limit (or a filtered limit) of rings is defined as follows.Suppose we're given a family of rings formula_88, "i" running over positive integers, say, and ring homomorphisms formula_163 such that formula_164 are all the identities and formula_165 is formula_166 whenever formula_167.Then formula_168 is the subring of formula_169 consisting of formula_170 such that formula_171 maps to formula_172 under formula_163.