Your question does not make sense. You do not "add your name as co-owner" of a trust. You can be the grantor of the trust, (which you are not, since this was your parents), the trustee, (which presumably is your mother), or the beneficiary. It sounds like you are beneficiary, or at least a contingent beneficiary, upon your mother's death. If you are asking if your being added as co-trustee, (or even sole trustee), would jeopardize the assets, then the answer is no. Your mother would still be the primary beneficiary, in all likelihood. Because you are talking about your mother's estate plan, however, she should be discussing these matters with her attorney.
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I agree with Mr. Frederick. There's no way to add your name as co-owner of the trust, nor do I see what the incentive would be to co-own anything of your mother's. Due to the foreclosure, you potentially face the risk of a deficiency judgment entered against you. A creditor could attach any assets you own to satisfy the judgment, including those you co-own with your mother. Perhaps you can provide some more clarity to your circumstance, but as it stands now, it's a fruitless venture.
I also agree with the other attorneys, I'm not sure what you mean by become a "co-owner" of the Trust. You need to make the distinction between whether you have a current interest in the money, or whether you are a beneficiary. If you are a beneficiary, you typically would not have a right to the money until your mother passes (or some other event depending on what the Trust says). If you are a beneficiary, creditors cannot take the asset right now because you have no current right to the asset. If you become an "owner", and have a current ownership interest or ability to access the funds for your personal use, creditors can potentially take that interest.
I am guessing that your intent here is to add you as a person legally entitled to help in the management of the assets of your mother's trust. For that you would, indeed, need to be a co-trustee. You can, however, acquire many of the same privileges of helping your mother to manage the assets via a durable power of attorney. Perhaps one was included with your parents' trust package, but it probably grants authority to manage assets only upon a finding of mental incapacity. That is called a "springing" power, when what you would need in this instance is an "immediate" power of attorney. This would not expose your mother's assets (in her name or that of her trust) to your creditors, as the assets would not change name until her passing . . . then, pursuant to the terms of the trust, they would be re-titled in your name.
I agree with my colleagues here that some proper additional planning may more effectively shelter those assets - both for the remainder of Mom's life and even after they become yours. You may, in the company of and with the cooperation of your mother, wish to consult legal counsel for further information.
Mr. Williams is licensed to practice law in the state of Nevada. The foregoing response does not constitute legal advice and does not create an attorney/client relationship. The response is, essentially, educational only, and is intended to provide general legal information about the matters presented by the question. Often, the question does not include significant and important facts, dates and other information that, if known, could significantly affect the appropriateness of the response and make it unsuitable. Mr. Williams strongly advises consulting directly with an attorney licensed in your state in order to ensure proper advice and counsel is received.