I was under the impression that Minoru could have lived had he been told about the rule and thus not taken the money. But since he told Ryuk not to contact him for any reason, Ryuk never told him the rule, and thus he died as soon as he got the money, as per the rule. Ryuk writing the name in the book was just him carrying out the punishment.
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Seplito Nash, Smelling Like the Vault since 1996 Step FOUR! Get Paid!