 The latter homomorphism is injective if "R" is a Noetherian integral domain and "I" is a proper ideal, or if "R" is a Noetherian local ring with maximal ideal "I", by Krull's intersection theorem.The construction is especially useful when "I" is a maximal ideal.The basic example is the completion Z"p" of Z at the principal ideal ("p") generated by a prime number "p"; it is called the ring of "p"-adic integers.