As above,, Eire is not part of Schengen. My wife has Schengen Residencia thus can travel to any Schengen country, she cannot go to UK, Gibraltar nor Eire
However, although UK is a 'pain the butt' when I called the Irish (southern) Government I was told as I am British we can make the application in the post and again because I am British my wife's visa would be free. Not 100 euros for a one entry to UK or 150 euros for Gib.