A Jack Russell will definitely do the job @Charltonstrek, but please don't get one as a rat catcher. However, if you have already had dogs, and actually want one at the moment, you could consider a Jack Russell, if, you want an energetic and fast speed dog, who loves to chase anything that is moving, particularly if it is moving fast! So he would try to chase anything from a rat - usually successfully - to a bicycle, or a car, or even a cheetah if he got the opportunity!
Many people will say that the dog's human, or if necessary, a professional trainer, would be able to train the chasing obsession out of a Jack Russell. I would answer that with a "No You Can't", and trying to do so would be a waste of the poor dog's time, and of the human's time as well. I am not saying that they can't be trained in any way, they can be trained not to wee or poop in the house, and they are usually very brave (stupid!) If they had to face 3 hungry wolves, they wouldn't turn and run away, they would run straight towards them, whilst barking at the top of their voices. I don't know what the wolves would do, and I would rather not think about it!
Treated properly, Jack Russells will usually adore their human/s, and be very loyal to them. They would also, almost certainly, 'guard' their human, which can cause a few problems in certain cases. If they take against someone, judging them as not being nice, they would consider that person to be a threat to themselves, and/or their family. J.R.'s are not quick to bite a person, they seem to prefer to intimidate them by barking very loudly, and showing them their big sharp teeth, but they can bite if they deem it necessary. I have never heard of a J.R. savaging someone - but I still wouldn't let him get too close to babies or young children, as children can be as unpredictable as dogs can, and maybe hurt the dog unintentionally.
I don't believe that Jack Russells are safe to be let off their lead anywhere, apart from their own home, and a secure garden. Another two places that would hopefully be secure enough to let them off the lead, would be in the middle of deserted countryside, or in a secure dog exercising field, one that you probably have to pay for! Anyone who has a Jack Russell 'should', in fact, 'must', make sure that they are taken frequently to safe places to run off their lead. If you think that all of that sounds great, and you love really cheeky dog's, then getting a Jack Russell could be the perfect dog for you OP.