AITA FOR IGNORING MY "FRIEND"

So I (13f) & my "friend"(13f) let's call her T are not on talking terms over this. T told my teacher(45f) let's call her Mrs. M that I told her to kll herself. So let's get this straight she said that during math the day before that I told her "Kll your self!" I was pulled in to the library and asked "Did you." I said no and that I didn't remember anything from the day before. Then Mrs.M pulled me in the library 5 minutes later. In there was Mrs.M and T, T was all red. Mrs.M asked her to tell me what I did. She started having a "panic attack". She didn't even say anything so Mrs.M had to tell me. Then Mrs.M told me she was going to call my mom then sent me back to my class crying. When I got home I was crying. Told my siblings and they all were pissed. My mom got home and I broke down a little kid cry and all. She was very pissed and then called my teacher. She explained that I couldn't have done. Mrs. M agreed and said there was holes in the story but didn't know I so torn up about it. To cheer me up we got ice cream and on the ride home guess who saw me T. She waved at us like a maniac. Then T tried to hang out with my sister but she ripped her a new one. I told my mom I didn't /can't go to school the next day. T acted cool the next day. Then the next week she asked why I was ignoring her I made up some lie and walked off so am I??