Are the distributive and abelian properties inherited from a ring to a SUBSET of the ring?(The subset is closed under both the operations)

Question

I was reading the proof of the theorem for proving subring and it assumed this. can someone verify.

the proof i was reading about(which i mentioned in the description) was about proving that a subset with some particular properties is a subring. it inferred from the fact that closure under both operations means it follows distributivity and also the commutative property under addition. I was confused how they inferred this directly, and that's what I asked in the question

Answer 1

"All of our cows are brown."

"What about the cows in that particular field, are they brown?"

"Yes, they are ours and they are brown."

Note that brown-ness and abielian-ness are properties that relate to individual elements (for brown-ness) or pairs of elements (for abielean-ness), and so they pass on almost automatically to subsets. It's good that you're worrying about this, as there are some properties that don't pass automatically to subsets. For example, "closure under inverses" does not.