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Reference.org
Algebra over a field
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, an algebra over a field (often simply called an algebra) is a <a href="/facts/Vector_space/M1pxMLx2">vector space</a> equipped with a <a href="/facts/Bilinear_map/LH1lHVeP">bilinear</a> <a href="/facts/Product_(mathematics)/aXY11nGS">product</a>. Thus, an algebra is an <a href="/facts/Algebraic_structure/TkuXca24">algebraic structure</a> consisting of a <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> together with operations of multiplication and addition and <a href="/facts/Scalar_multiplication/jroFDfbM">scalar multiplication</a> by elements of a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> and satisfying the axioms implied by "vector space" and "bilinear". </p><p>The multiplication operation in an algebra may or may not be <a href="/facts/Associative/EQX8lV6r">associative</a>, leading to the notions of <a href="/facts/Associative_algebra/DRfoF6KV">associative algebras</a> where associativity of multiplication is assumed, and <a href="/facts/Non-associative_algebra/Vk9Kjmrs">non-associative algebras</a>, where associativity is not assumed (but not excluded, either). Given an integer <i>n</i>, the <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> of <a href="/facts/Real_matrix/qa8FI4ko">real</a> <a href="/facts/Square_matrix/5TP9mNNn">square matrices</a> of order <i>n</i> is an example of an associative algebra over the field of <a href="/facts/Real_number/R02gw5Pb">real numbers</a> under <a href="/facts/Matrix_addition/6zzLWyL0">matrix addition</a> and <a href="/facts/Matrix_multiplication/lYDB6Ro2">matrix multiplication</a> since matrix multiplication is associative. Three-dimensional <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a> with multiplication given by the <a href="/facts/Vector_cross_product/uRBJzkcW">vector cross product</a> is an example of a nonassociative algebra over the field of real numbers since the vector cross product is nonassociative, satisfying the <a href="/facts/Jacobi_identity/C9HoyQ3z">Jacobi identity</a> instead. </p><p>An algebra is unital or unitary if it has an <a href="/facts/Identity_element/9nss3xHY">identity element</a> with respect to the multiplication. The ring of real square matrices of order <i>n</i> forms a unital algebra since the <a href="/facts/Identity_matrix/Glbcf6ol">identity matrix</a> of order <i>n</i> is the identity element with respect to matrix multiplication. It is an example of a unital associative algebra, a <a href="/facts/Unital_ring/JnCUXz63">(unital) ring</a> that is also a vector space. </p><p>Many authors use the term <i>algebra</i> to mean <i>associative algebra</i>, or <i>unital associative algebra</i>, or in some subjects such as <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>, <i>unital associative commutative algebra</i>. </p><p>Replacing the field of scalars by a <a href="/facts/Commutative_ring/QS1r2ovR">commutative ring</a> leads to the more general notion of an algebra over a ring. Algebras are not to be confused with vector spaces equipped with a <a href="/facts/Bilinear_form/gyvzn9ym">bilinear form</a>, like <a href="/facts/Inner_product_space/HoyjElEy">inner product spaces</a>, as, for such a space, the result of a product is not in the space, but rather in the field of coefficients. </p>