## Which of the following ring is not an Artinian ring?

commutative Noetherian ring

The ring of integers is a commutative Noetherian ring but is not Artinian. This means that not all prime ideals in Z are maximal.

## Is Artinian ring commutative?

Commutative Artinian rings A is a finite product of commutative Artinian local rings. A / nil(A) is a semisimple ring, where nil(A) is the nilradical of A. A has Krull dimension zero. (In particular, the nilradical is the Jacobson radical since prime ideals are maximal.)

**Are Artinian modules finitely generated?**

Since an Artinian ring is also a Noetherian ring, and finitely-generated modules over a Noetherian ring are Noetherian, it is true that for an Artinian ring R, any finitely-generated R-module is both Noetherian and Artinian, and is said to be of finite length; however, if R is not Artinian, or if M is not finitely …

**Why is Z Noetherian?**

Z[√2] is a finitely generated Z-module, so it is noetherian as a Z-module. This implies it is a noetherian ring, since every (left) ideal of Z[√2] is also a Z-module.

### What is the definition of an Artinian ring?

Definition 10.52.1. A ring is Artinian if it satisfies the descending chain condition for ideals. Lemma 10.52.2. Suppose is a finite dimensional algebra over a field. Then is Artinian. Proof. The descending chain condition for ideals obviously holds. Lemma 10.52.3. If is Artinian then has only finitely many maximal ideals. Proof.

### How is an Artinian ring named after Emil Artin?

In abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes…

**Is the ring of matrices left Artinian or right Artinian?**

The Artin–Wedderburn theorem characterizes all simple Artinian rings as the ring of matrices over a division ring. This implies that a simple ring is left Artinian if and only if it is right Artinian.

**Which is locally nilpotent in an Artinian ring?**

Any ring with finitely many maximal ideals and locally nilpotent Jacobson radical is the product of its localizations at its maximal ideals. Also, all primes are maximal. Proof. Let be a ring with finitely many maximal ideals . Let be the Jacobson radical of . Assume is locally nilpotent. Let be a prime ideal of .