What about in a ring where we have two binary operations defined. I get super confused when I see someone just switch from something like n*a to a + a + ... + a (n times, where * is the binary operation on the "semigroup" part of the ring, and + is the operation on the "group" part of the ring) because I freak out and think, how do I know these two things produce the same result? I mean I get it for familiar number systems like R and C and Z, but does it always work out? I just feel uneasy with generalizing it to everything! Can I? Is there a theorem for that?