We have the idea of a prime ideal in a commutative ring $R$ but in universal algebra, we generalize the notion of ideal to that of a congruence. I’ve thought over the question of what a prime congruence would look like and all I could say is that maximal (proper) congruences should be prime and the trivial congruence shouldn’t be. Given the importance of prime ideals, I’d imagine someone’s put thought into how this would generalize though I haven’t been able to find anything.

I ask this question because I’m thinking about that the spectrum of a (commutative?) algebraic structure would look like in general. Any pointers would be helpful.

(Should I put the AG tag here?)