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Ridley was out-of-contract at Glanford Park, and had been offered a deal by League Two side Bradford, as well as Cheltenham and a new two-year contract at United.
Lincoln City were also linked with a move for the defender, who has made the move to guarantee regular first-team football.
"I'm really pleased that Lee has now signed for us," boss John Ward told the Robins' official website.
"The move was held up by our respective holidays but now that the details have been finalised I think it shows the commitment of both the club and the player that we have agreed a three year contract.
"Lee is a young player who has made over 100 Football League appearances. He has played against us for Scunthorpe and we have seen him in action on a number of other occasions as well.
"I spoke to Brian Laws [Sheffield Wednesday manager] who was previously at Scunthorpe for many years and he gave Lee a very high recommendation."