Which of the following ring is not an Artinian ring?

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 .