Nonassociative rings and algebras Definition

Nonassociative algebra

Builds on Commutative ring Module over a ring Nonassociative ring

Definition

Let $R$ be a commutative ring with unit. A nonassociative algebra over $R$ is an $R$ -module $A$ with a map $A \times A \to A$ , $(x, y) \mapsto xy$ , that is $R$ -bilinear: $(rx + sy)z = r(xz) + s(yz) \quad \text{and} \quad z(rx + sy) = r(zx) + s(zy)$ for all $r, s \in R$ and $x, y, z \in A$ . Neither associativity nor a unit element is required.

Remarks

With $R = \Z$ , a nonassociative algebra is the same as a nonassociative ring. When $R$ is a field, it is an algebra over a field. Example: $\R^3$ with the cross product is a nonassociative algebra over $\R$ , because the cross product is bilinear. It is not associative: $(e_1 \times e_1) \times e_2 = 0$ , while $e_1 \times (e_1 \times e_2) = e_1 \times e_3 = -e_2$ . Non-example: $\R$ with the product $x * y = x + y$ is not an algebra, because $(2x) * y = 2x + y$ differs from $2(x * y) = 2x + 2y$ when $y \neq 0$ .

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Tag	`QS36GTKQAZ`
URI	`m2://QS36GTKQAZ`
Kind	definition
Writing	W0 stub
MSC	17A01
Defines	Nonassociative algebra
Uses	Commutative ring, Module over a ring, Nonassociative ring
Changed	3 November 2026

Nonassociative algebra

Definition

Remarks

Reading path

Further reading

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