- algebra, higher algebra
- universal algebra
- monoid, semigroup, quasigroup
- nonassociative algebra
- associative unital algebra
- commutative algebra
- Lie algebra, Jordan algebra
- Leibniz algebra, pre-Lie algebra
- Poisson algebra, Frobenius algebra
- lattice, frame, quantale
- Boolean ring, Heyting algebra
- commutator, center
- monad, comonad
- distributive law

- group, normal subgroup
- action, Cayley's theorem
- centralizer, normalizer
- abelian group, cyclic group
- group extension, Galois extension
- algebraic group, formal group
- Lie group, quantum group

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

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 $R$ with operations of addition and multiplication, such that:

- $R$ is a unital magma under multiplication;
- $R$ 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 $R$, a non-associative commutative ring extension (in the sense of field extension) of $R$ is just a $R$-nonassociative algebra.

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

Last revised on August 21, 2024 at 02:37:58. See the history of this page for a list of all contributions to it.