Groups/Rings/Fields question on Elementary Number Theory

Question

So I understand that for a set to be a GROUP, it must follow certain properties(closure, association etc.). But what I don't get is how do I know during validation of these properties if i must use addition or multiplication for example (a+b=x) or (a.b=x) to prove these properties

Answer 1

It ought to be clear from context. In introductory group theory, groups are typically introduced as $(G,\ast)$, so the choice of $\ast$ makes it clear what the binary operation is.

For instance, you may consider $G = (\mathbb{Z},+)$, or $H = (\mathbb{Z},\times)$.

If you were to just see $G = \mathbb{Z}$, focus on the context, and reread the passage until you are clear which operation you are dealing with.

Answer 2

For a group there's only one operation, and the operation should basically be specified to check if it's a group. A field and ring has 2 operations, $+$ and $\cdot$. And in the definition it's specified for both operations which rules they should obey. https://en.wikipedia.org/wiki/Ring_(mathematics)#Definition