Can a cut/wound be infected if there is no type of drainage but it has a ring of red around it and is sore?

Yes: Signs of inflamation like , surruonding redness, swelling, tenderness can indicate infection, even without drainage.

See below: Anything is possible, so to play it safe have a doctor look at you. God luck from nyc.

Yes: Initially you might not see drainage but the soreness and the red borders can alert you that the infection is developing. I would start using triple antibiotic ointment and cover the cut with a bandaid until it heals completely and if the redness area continues to grow despite the treatment seek medical attention.