# Thread: rings and ideals

1. ## rings and ideals

Let $\displaystyle I_{1} \subseteq I_{2} \subseteq I_{3} \ldots$ be an increasing sequence of ideals in a ring R

1.) Prove that $\displaystyle \bigcup_{i\in\aleph} I_{i}$ is an ideal.

2.) Prove that if every ideals of R is finitely generated, then exist a number $\displaystyle n \in \aleph$ such that
$\displaystyle I_{1} \subseteq I_{2} \ldots \subseteq I_{n-1}\subseteq I_{n} = I_{n} = I_{n+1} =I_{n+2} = \ldots$

2. Originally Posted by Magics6
Let $\displaystyle I_{1} \subseteq I_{2} \subseteq I_{3} \ldots$ be an increasing sequence of ideals in a ring R

1.) Prove that $\displaystyle \bigcup_{i\in\aleph} I_{i}$ is an ideal.

2.) Prove that if every ideals of R is finite, then exist a number $\displaystyle n \in \aleph$ such that
$\displaystyle I_{1} \subseteq I_{2} \ldots \subseteq I_{n-1}\subseteq I_{n} = I_{n} = I_{n+1} =I_{n+2} = \ldots$

Use definitions, as simple as that...and in (2), shouldn't it be "finitely generated ideals" instead of "finite ideals"? Because if it is finite then it is a very trivial exercise.

Tonio

3. Originally Posted by tonio
...and in (2), shouldn't it be "finitely generated ideals" instead of "finite ideals"?
confirm... finitely generated ideals

4. Originally Posted by Magics6
Let $\displaystyle I_{1} \subseteq I_{2} \subseteq I_{3} \ldots$ be an increasing sequence of ideals in a ring R

1.) Prove that $\displaystyle \bigcup_{i\in\aleph} I_{i}$ is an ideal.
Let $\displaystyle I= \bigcup_{i\in\aleph} I_{i}$; let $\displaystyle x, y \in I$ and $\displaystyle r \in R$. Without loss of generality, $\displaystyle x \in I_n$ and $\displaystyle y \in I_m$. Let N=max{m,n}.

Since $\displaystyle I_N$ is an ideal in R, we see that $\displaystyle x + y \in I_N \subset I$, $\displaystyle rx \in I_N \subset I$ and $\displaystyle xr \in I_N \subset I$. Thus I is an ideal in R.

2.) Prove that if every ideals of R is finitely generated, then exist a number $\displaystyle n \in \aleph$ such that
$\displaystyle I_{1} \subseteq I_{2} \ldots \subseteq I_{n-1}\subseteq I_{n} = I_{n} = I_{n+1} =I_{n+2} = \ldots$
Assume every ideal in R is finitely generated and consider I in (1). It follows that I is finitely generated by $\displaystyle a_1, a_2, \cdots, a_m$. Each $\displaystyle a_i$ for i=1,2,...,m lies in one of the chains of I, say $\displaystyle I_{k_i}$. Let $\displaystyle n= \text{max}\{k_1, k_2, \cdots, k_m\}$. Then $\displaystyle a_i \in I_n$ for all i. This implies $\displaystyle I \subseteq I_n$. Thus, $\displaystyle I_n =I$ and $\displaystyle I_k = I_n$ for all $\displaystyle k \geq n$.