New Jersey is an "equitable property" state. This means that all marital property acquired during the marriage should be divided equally, unless the court finds that equal division would be unjust. This is not really a tax question, but a family law question. You need to see a family law attorney to determine your marital rights.
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You are in an equitable, rather than a community property state. This is a family law rather than a tax question and you should see a family law attorney to address this issue. Also repost this question to the family law area.
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She may be entitled to half if there is equitable distribution during a divorce. Although, as you claim, she "did not work", chances are that she provided services by taking care of the children (if applicable) or the house. As such, her entitlement to the said funds is not necessarily inappropriate.
The debt can be legally satisfied from the income tax return.
Each case is different and you are advised to seek personal counseling with an attorney in order to get more detailed advice.
Both the tax refund and the debt are joint property subject to equitable distribution. So, yes, your wife is entitled to a share of the refund and is also jointly responsible for the debt. Once you are having these sorts of disputes it's probably in your interest to file for divorce sooner rather than later.
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