nLab nonassociative ring




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.



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

  • RR is a unital magma under multiplication;
  • RR is an abelian group under addition;
  • multiplication distributes over addition.

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


A nonassociative ring is a unital magma object in Ab.


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.


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


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

Last revised on October 4, 2023 at 17:16:06. See the history of this page for a list of all contributions to it.