Brain Ventricular Enlargement
Definition
$N$ is referred to as a ring.
- When the ideals of $N$ satisfy $S_{1} \le S_{2} \le \cdots$, it is called an Ascending Chain.
- For the ascending chain $\left\{ S_{i} \right\}_{i \in \mathbb{N} }$, if there exists $n \in \mathbb{n}$ that satisfies $S_{n} = S_{n+1} = \cdots$, it is said to be Stationary. In other words, in a stationary ascending chain, the ideal does not increase any further from a certain point onwards.
- A ring in which all ascending chains are stationary is called a Noetherian ring.
- Conversely, for chains that decrease gradually, the term Descending is used.
Explanation
Chains are not a concept exclusive to ideals, and in set theory, they are dealt with much more rigorously and generally by defining partially ordered sets or relations. However, the place where ascending chains are usually needed is in algebra, and there is no need to understand it in such a complicated way in algebra.
The existence of the largest thing in a certain repetitive structure within a finite place is not as obvious as it might seem. Therefore, the condition of being a Noetherian ring is a very favorable one, and the following famous theorem has been applied in various fields.
Hilbert’s Basis Theorem
If $N$ is a Noetherian ring, then $N [ x_{1} , \cdots , x_{n} ]$ is also a Noetherian ring.
See Also
- Euclidean domain $\implies$ Principal ideal domain $\implies$ Noetherian ring