Mather said he was departing in order to try to restore calm to the club, two days after former director Paul Murray won a courtroom victory to force a postponement of next week's schedule annual general meeting in a dispute over a possible election to determine whether he and his allies should join the board.
The club also announced, via a statement to the stock exchange, the resignation of non-executive director Bryan Smart.
In a statement on Rangers' website, Mather said: "The interests of the club are of paramount importance and I believe these are best served by me leaving the club."
Mather went on to say he regretted the ongoing boardroom battles, but insisted he had left the club on a solid footing despite the challenges posed.
He said: "Despite recent events and speculation, the facts of the matter are that the Club is financially secure and in a far better place than it was a year ago.
"Unlike most football clubs Rangers has money in the bank, no borrowings and this season we have assembled a squad which is capable of progressing through the leagues.
"I have enjoyed a very constructive relationship with Ally McCoist and wish him and the team every success.
"My short tenure as chief executive has been beset by incessant attempts to destabilise the operations of the club, all done supposedly in the interests of Rangers.
"I had real faith in the rebuilding of Rangers and invested significantly in the club. Sadly, those who have been most active in upsetting the very good progress we have been making were not willing to do the same.
"I leave with my head held high and will remain as a shareholder and a supporter of Ally and his team."