Question

If $\left[I_k\right]$ is a (possibly infinite) family of ideals in R, prove that the intersection of all the role="math" localid="1649753314246" ${\mathit{I}}_{\mathit{k}}$is an ideal.

Short answer

It is proved ${\cap }_k{I}_k$is an ideal in R.

Step-by-step solution

Theorem

Let R be a ring, and Ibe a non-empty subset of the ring. Then, I is said to be the ideal if and only if it has the following two properties:

1. If $a\mathit{,}b\mathit{\in }I$, then $a-b\in I$.
2. If $r\mathit{\in }R$and $a\in I$, then $ra\in I$, and $ar\in I$.

Proof part

It is given that $I_k$are ideals and all are non-empty, so using the theorem stated in step 1, ${\cap }_k{I}_k$is non-empty.

Now, assume that role="math" localid="1649753631365" $a,b\in {\cap }_k{I}_k$; then, using the definition of intersection, we have $a,b\in I_k$.

Since $I_k$ are ideals, $a-b\in I_k$ for all k, which implies $a-b\in {\cap }_k{I}_k$. Thus, property (i) of the theorem is satisfied.

Next, assume that $r\in R$.

As $a\in I_k$and Iare ideal, we have $ra\in I_k$ and $ar\in I_k$. $ar\in I_k$.

By definition, we have $ra\in {\cap }_k{I}_k$and $ar\in {\cap }_k{I}_k$, so property (ii) of the theorem is satisfied.

Hence, it is proved that ${\cap }_k{I}_k$is an ideal in R.