nLab nonassociative ring

Contents

Context

Algebra

higher algebra

universal algebra

Contents

Idea

The concept of nonassociative ring is like that of ring but without the requirement of associativity.

Remark on terminology

The term “nonassociative ring” may be regarded as an example of the “red herring principle”, as a nonassociative ring is not in general a ring in the modern sense of the word.

Definition

Definition

A nonassociative ring is a set $R$ with operations of addition and multiplication, such that:

• $R$ is a unital magma under multiplication;
• $R$ is an abelian group under addition;

More sophisticatedly, just as a ring is a monoid object in Ab, so

Definition

A nonassociative ring is a unital magma object in Ab.

Remark

A non-associative ring may well have associativiy, i.e. it may be in the image of the forgetful functor from associative rings to nonassociative rings. But if so, then this element is still not part of the defining data and in particular a homomorphism of non-associative rings need not to preserve associativity.

Properties

For any nonassociative commutative ring $R$, a non-associative commutative ring extension (in the sense of field extension) of $R$ is just a $R$-nonassociative algebra.

Examples

Examples include Lie rings and nonassociative algebras such as alternative algebras.

Last revised on May 3, 2021 at 15:21:19. See the history of this page for a list of all contributions to it.