Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms implied by "vector space" and "bilinear".
Three-dimensional Euclidean space with multiplication given by the vector cross product is an example of a nonassociative algebra over the field of real numbers since the vector cross product is nonassociative, satisfying the Jacobi identity instead.
An algebra is unital or unitary if it has an identity element with respect to the multiplication.
The ring of real square matrices of order n forms a unital algebra since the identity matrix of order n is the identity element with respect to matrix multiplication.
In symbols, we say that a subset L of a K-algebra A is a subalgebra if for every x, y in L and c in K, we have that x · y, x + y, and cx are all in L. In the above example of the complex numbers viewed as a two-dimensional algebra over the real numbers, the one-dimensional real line is a subalgebra.
These types are specified by insisting on some further axioms, such as commutativity or associativity of the multiplication operation, which are not required in the broad definition of an algebra.
One may define a unital zero algebra by taking the direct sum of modules of a field (or more generally a ring) K and a K-vector space (or module) V, and defining the product of every pair of elements of V to be zero.
If e1, ... ed is a basis of V, the unital zero algebra is the quotient of the polynomial ring K[E1, ..., En] by the ideal generated by the EiEj for every pair (i, j).
For example, the theory of Gröbner bases was introduced by Bruno Buchberger for ideals in a polynomial ring R = K[x1, ..., xn] over a field.
Examples detailed in the main article include: The definition of an associative K-algebra with unit is also frequently given in an alternative way.
This definition is equivalent to that above, with scalar multiplication given by Given two such associative unital K-algebras A and B, a unital K-algebra homomorphism f: A → B is a ring homomorphism that commutes with the scalar multiplication defined by η, which one may write as for all
In other words, the following diagram commutes: For algebras over a field, the bilinear multiplication from A × A to A is completely determined by the multiplication of basis elements of A. Conversely, once a basis for A has been chosen, the products of basis elements can be set arbitrarily, and then extended in a unique way to a bilinear operator on A, i.e., so the resulting multiplication satisfies the algebra laws.
Thus, given the field K, any finite-dimensional algebra can be specified up to isomorphism by giving its dimension (say n), and specifying n3 structure coefficients ci,j,k, which are scalars.
These structure coefficients determine the multiplication in A via the following rule: where e1,...,en form a basis of A.
Note however that several different sets of structure coefficients can give rise to isomorphic algebras.
Thus, the structure coefficients are often written ci,jk, and their defining rule is written using the Einstein notation as If you apply this to vectors written in index notation, then this becomes If K is only a commutative ring and not a field, then the same process works if A is a free module over K. If it isn't, then the multiplication is still completely determined by its action on a set that spans A; however, the structure constants can't be specified arbitrarily in this case, and knowing only the structure constants does not specify the algebra up to isomorphism.
Two-dimensional, three-dimensional and four-dimensional unital associative algebras over the field of complex numbers were completely classified up to isomorphism by Eduard Study.
According to the definition of an identity element, It remains to specify There exist five such three-dimensional algebras.
Taking into account the definition of an identity element, it is sufficient to specify The fourth of these algebras is non-commutative, and the others are commutative.
In some areas of mathematics, such as commutative algebra, it is common to consider the more general concept of an algebra over a ring, where a commutative ring R replaces the field K. The only part of the definition that changes is that A is assumed to be an R-module (instead of a K-vector space).
[6] On the other hand, not all rings can be given the structure of an algebra over a field (for example the integers).