First, a comment on terminology: If a person dies leaving a will, the legal procedure for the appointment of the executor is called a probate proceeding or an application for issuance of letters testamentary. If, however, there was no will but there is an estate that has to be administered, then the legal procedure is called an application for issuance of letters of administration, and the law governing the distribution of the estate is known as the law of intestacy.
You indicated in your question that your husband had no will. In that case, if he owned property (such as real estate, bank accounts, stocks, bonds etc.), then you (or his children if he had any) would have to obtain letters of administration from the Surrogate's Court in order to administer his estate, including paying his debts from estate assets. You are correct that where there are no assets in his name, there is nothing to administer and administration proceedings are not necessary.
You are not personally liable for your husband's debts unless the debts were incurred by him in purchasing items necessary for your health, comfort and welfare, such as groceries or medicine. In those cases, you would be responsible, not your husband's estate.
Under the circumstances described in your question, you do not have to do anything. That does not prevent one of the creditors from asking the Surrogate's Court to appoint him or her as administrator of your husband's estate. Still, if your husband owned nothing, then the creditor's rights are not enhanced by getting himself appointed as administrator of the estate.
-- Richard A. Whitney
