# Is the Intersection of Subrings of R a Subring of R?

• lostinmath08
In summary, the question is asking to show that if S and T are subrings of R, then their intersection is also a subring of R. The requirements for a subset of R to be a subring are that it must satisfy certain axioms, such as being closed under addition and multiplication. The subring of R generated by a subset X is the smallest subring of R containing X.
lostinmath08
1. The Problem

If S and T are subrings of a ring R, show that S intersects T, is a subring of R.

## The Attempt at a Solution

What are the requirements for a subset of R to be a subring?

The following axioms must be satisfied
a) (for all or any) x,y E R implies x+(-y) E R
b) (for all or any) x,y E R implies xy E R ( R is closed under mulitplication)

The above are the requirements for a subring to be valid.

This is something i got from wikipedia:

Let R be a ring. Any intersection of subrings of R is again a subring of R. Therefore, if X is any subset of R, the intersection of all subrings of R containing X is a subring S of R. S is the smallest subring of R containing X. ("Smallest" means that if T is any other subring of R containing X, then S is contained in T.) S is said to be the subring of R generated by X. If S = R, we may say that the ring R is generated by X.

So then does S=T?

## 1. What is a subring?

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.

## 2. How do you prove that S∩T is a subring of R?

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.

## 3. What is the significance of proving that S∩T is a subring 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.

## 4. Can S∩T be a subring of R if S and T are not subrings of R on their own?

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.

## 5. What are some common mistakes to avoid when proving S∩T is a subring of R?

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.

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