Nonassociative rings and algebras

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• ConceptNonassociative ring

  A nonassociative ring is an abelian group with a multiplication that distributes over addition on both sides, without the requirement that it be associative. "Nonassociative" means "not necessarily associative", so associative rings are a special case. Lie algebras, Jordan algebras and the octonions are the main examples where associativity fails. A nonassociative algebra over a field is the same notion with a compatible vector space structure.

• ConceptMagma

  A magma is a set with one binary operation and no further axioms. It is the most basic algebraic structure with a binary operation: semigroups, monoids, groups and quasigroups are magmas with extra conditions. Older texts call a magma a groupoid, a name that now usually means something else in category theory.

• ConceptDerivation

  A derivation is a linear map on an algebra that satisfies the product rule $D(ab) = D(a) b + a D(b)$, see the definition. It is the algebraic form of the derivative, and it makes sense without limits. Derivations are basic in differential algebra, algebraic geometry and Lie theory. The derivations of a Lie algebra obey the same rule for the Lie bracket. The commutator of two derivations is again a derivation, so the derivations of an algebra form a Lie algebra.

• ConceptSimple algebra

  A simple algebra is a nontrivial algebra whose only congruences are the two trivial ones. So it has no proper nontrivial quotient algebras: every homomorphism out of it is injective or constant. The notion covers simple groups and simple rings, and it is the basic building block in the study of varieties.

• ConceptFree algebra

  The free algebra $k\langle X \rangle$ on a set $X$ over a commutative ring $k$ is the algebra of polynomials in noncommuting variables from $X$. It is the free object among associative algebras over $k$: any map from $X$ into an algebra extends uniquely to an algebra homomorphism. So every algebra generated by a set $X$ is a quotient of $k\langle X \rangle$, and algebras are often given by generators and relations in this way. It is the noncommutative analogue of the polynomial ring and is isomorphic to the tensor algebra of the free module with basis $X$.

• ConceptVariety of algebras

  A variety of algebras is the class of all algebras of a fixed signature that satisfy a given set of identities. Groups, rings, lattices and modules over a fixed ring are the standard examples. Varieties are the basic objects of universal algebra, because many constructions, such as free algebras and presentations, work in every variety at once. By Birkhoff's theorem, a class of algebras is a variety exactly when it is closed under homomorphic images, subalgebras and direct products.