<human>: Give me the gist of the text in a nutshell

Hey there, I could really use some outside perspective as I'm really torn on this issue. I need someone to talk to and give me unbiased opinions :)

First off, when I first starting dating my bf, I found out he was cheating on me very early on after we verbally established exclusivity. He apologized and we moved on. He cut off contact with the other woman. He was still communicating with his ex-girlfriend and it was pretty obvious to me he was keeping her as a backup during this time. His ex is not a very nice person: while we were dating, she was actively making fun of me (saying how I am ugly multiple times), using racial slurs, etc. He did not defend me so I finally gave him an ultimatum that he needed to send her an email that they need to stop communicating or I'm out. He did exactly that and showed me the email. He seemed very remorseful that he hurt me and since then he has been a very good boyfriend. 

However, this is where I need your opinions. Recently his ex's mother died, and a mutual friend of my bf and his ex contacted him to tell him to contact the ex and talk to her. He did not communicate with her but he did send flowers to her and the family. I guess the ex became bitter that he didn't bother to call her so she sent him a message stating that 'she's sad things had to turn out like this (the no communication) but she appreciates the flowers.'

I know it was very nice of my bf to send the ex flowers because her mom died, but something about this rubs me the wrong way. In the back of my mind, I'm wondering if he did this to keep her in the background just in case something happens between us so he can use this to get back with her. I know that he didn't technically break his promise with me but something just seems off.

There are details that I left out but I didn't want to bore everyone with paragraphs. Thank you for reading this.
<bot>: bf sent his ex flowers while he agreed to sever ties to her because of past cheating and her being a bitch to me unprovoked. I'm torn. Should I let this go?