Let R be a commutative ring. Show that R / N(⟨0⟩) has no nonzero nilpotent elements.

Name: Problem 51, Chapter 14: Contemporary Abstract Algebra
Uploaded: 2020-05-20T17:26:08-08:00
Duration: 3 min 19 s
Channel: Joe Lesueur
Description: Let R be a commutative ring. Show that R / N(⟨0⟩) has no nonzero nilpotent elements.

step 1 And this problem, we must prove that matrix A is nil potent by raising it to sum power P and res

Let R be a commutative ring. Show that R / N(⟨0⟩) has no...

Key Concepts

Nilradical

The nilradical of a ring is the set of all nilpotent elements in that ring. It forms an ideal, and it is the largest ideal of the ring in which every element is nilpotent. The nilradical plays an important role in understanding the structure of a ring, as factoring it out often simplifies the ring by removing 'bad behavior' associated with nilpotent elements.

Quotient Ring

A quotient ring is formed by taking a ring and partitioning its elements into cosets determined by an ideal. The operations of addition and multiplication in the quotient ring are defined in a natural way relative to the operations in the original ring. This construction is widely used to study ring properties by 'modding out' certain undesirable elements or to simplify the analysis of a ring's structure.

Commutative Ring

A commutative ring is an algebraic structure consisting of a set equipped with two binary operations (addition and multiplication) where the addition forms an abelian group, the multiplication is associative, and multiplication distributes over addition. Moreover, in a commutative ring the multiplication is commutative. This structure is fundamental in algebra because many properties, like ideals and quotient structures, have simpler behavior when the ring is commutative.

Nilpotent Element

A nilpotent element in a ring is an element 'a' for which there exists some positive integer n such that a^n = 0. In ring theory, the behavior of nilpotent elements helps identify and characterize certain subsets of the ring, especially in connection with the notion of the radical of an ideal.

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