Desire to be right always

Hi coaches,
The more I put myself out there (by taking leadership at work) the more I find myself having the desire to be right all the time. I cannot accept my peer challenging me ( I have a single peer who I look at as my competition and he is critical of my work )
I wish there are no conflicts and everything is easy, but it’s not. I get very uncomfortable if he/somebody would find flaws in my project or my decision and tell me I could have done better. I make it mean that I’m inadequate.

How do I disassociate myself from my work?
How can I make this truly fun for myself?