A subring is a subset of a ring that is also a ring, meaning it contains the same operations of addition and multiplication, and follows the same rules, such as closure and associativity.
To prove that S∩T is a subring of R, you must show that it satisfies the three conditions of a subring: it is closed under addition and multiplication, and it contains the additive and multiplicative identities of R.
Proving that S∩T is a subring of R is important because it allows us to use all the properties and theorems of rings on S∩T, making it easier to analyze and solve problems involving this subset.
Yes, it is possible for S∩T to be a subring of R even if S and T are not subrings of R individually. This is because the intersection of two subrings may still satisfy the three conditions of a subring.
Some common mistakes to avoid when proving S∩T is a subring of R include assuming that S and T are subrings without checking all the conditions, not showing closure under addition and multiplication, and forgetting to include the additive and multiplicative identities of R in S∩T.