Under what terms are a no compete agreement valid?

Non-compete agreements are generally more closely scrutinized, and held to be unenforceable, than most contracts, but (except in California) they will generally be enforced if reasonably necessary to protect the business of the employer (most non-competes are exectued in the employment context; if this wasn't, it might change the analysis somewhat).  Whether the protection is reasonable depends on too many factors to try to analyize based on your short question, particularly as I am not a Wisconsin lawyer.  Suffice it to say that most non-competes will expressly prohibit the scheme you propose, i.e. for your wife to compete in a business you own, anad you should check the agreement in question to see if it does.  Even if it doesn't expressly prohibit this ruse, it is doubtful that a court will allow your wife to circumvent a valid non-compete in this manner.  In other words, the non-compete may not be enforceable, depending on circumstances, but if it is, the fact that the competing business is titled in your name rather than your wife's will probably not help you.