Is it ok if I extract the broken tooth #15 and not doing the implant? (photo)

You will need to extract the broken tooth # 15 as it can be a potential source of infection or if there is decay it can potentially spread to adjacent teeth. Your choice of not doing an implant is acceptable; however, be aware that in some individuals and in some instances the opposing bottom tooth may rise up into this empty space over time. Similarly the tooth adjacent to # 15 can migrate into this space in some cases. But in most instances not having an implant in this area would allow you to function just fine.

Since your second molar looks in a bad shape, it is better to get it removed. Next step should be tooth replacement. Fixed bridge is not possible as there is no tooth behind the first molar which can be used to derive support from. Dental implant placement is the only recommended solution  for you. Leaving it unreplaced can cause the lower molar to grow up (supraerupt). This can disturb your entire bite pattern. Besides, molars are needed for chewing food as well.
Hence, I would advise you to go for dental implant placement after you get your second molar removed.

Hello and thank you for your question, Vincent. I would suggest to remove the broken tooth in order to prevent possible infection. In regards to replacing that tooth, if it would be me, I would replace it with an implant.All the best to you!