Under the revised UCC, there is a section which deals with "accord and satisfaction".
http://www.law.cornell.edu/ucc/3/3-311.html
It probably would not apply in this circumstance because the amount of the claim must be unliquidated -- meaning the exact amount owed is not quantified. The amount tendered must be tendered in good faith, the amount of the claim must be unliquidated, and the person who receives payment must actually cash the check. Even then, they have a period of time to change their mind and tender a refund in order to avoid accord and satisfaction.
This section is intended for situations like where Jane slips and falls at Robert's house and gets injured. Robert (or his insurer) probably owes Jane something, but how much is subject to debate. If Robert makes a proposed settlement by tendering a check to her with the notation full payment -- and it is in an amount that he in good faith believes is sufficient -- and she accepts the payment and doesn't subsequently seek a refund, there is deemed to be an accord and satisfaction.