Editing Algebra over a field (section)

=== Definition ===
Let {{mvar|K}} be a [[field (mathematics)|field]], and let {{mvar|A}} be a [[vector space]] over {{mvar|K}} equipped with an additional [[binary operation]] from {{math|''A'' × ''A''}} to {{mvar|A}}, denoted here by {{math|·}} (that is, if {{mvar|x}} and {{mvar|y}} are any two elements of {{mvar|A}}, then {{math|''x'' · ''y''}} is an element of {{mvar|A}} that is called the ''product'' of {{mvar|x}} and {{mvar|y}}).  Then {{mvar|A}} is an ''algebra'' over {{mvar|K}} if the following identities hold for all elements {{math|''x'', ''y'', ''z''}} in {{mvar|A}} , and all elements (often called [[scalar (mathematics)|scalar]]s) {{mvar|a}} and {{mvar|b}} in {{mvar|K}}:
* Right [[distributivity]]: {{math|1=(''x'' + ''y'') · ''z'' = ''x'' · ''z'' + ''y'' · ''z''}}
* Left distributivity: {{math|1=''z'' · (''x'' + ''y'') = ''z'' · ''x'' + ''z'' · ''y''}}
* Compatibility with scalars: {{math|1=(''ax'') · (''by'') = (''ab'') (''x'' · ''y'')}}.

These three axioms are another way of saying that the binary operation is [[bilinear operator|bilinear]]. An algebra over {{mvar|K}} is sometimes also called a ''{{mvar|K}}-algebra'', and {{mvar|K}} is called the ''base field'' of {{mvar|A}}. The binary operation is often referred to as ''multiplication'' in {{mvar|A}}. The convention adopted in this article is that multiplication of elements of an algebra is not necessarily [[associativity|associative]], although some authors use the term ''algebra'' to refer to an [[associative algebra]].

When a binary operation on a vector space is [[commutative]], left distributivity and right distributivity are equivalent, and, in this case, only one distributivity requires a proof. In general, for non-commutative operations left distributivity and right distributivity are not equivalent, and require separate proofs.