Editing Algebra over a field (section)

== Kinds of algebras and examples ==

Algebras over fields come in many different types. 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. The theories corresponding to the different types of algebras are often very different.

=== Unital algebra ===

An algebra is ''unital'' or ''unitary'' if it has a [[Unit (algebra)|unit]] or identity element ''I'' with ''Ix'' = ''x'' = ''xI'' for all ''x'' in the algebra.

=== Zero algebra ===

<!--The term '''zero algebra''' may have other uses outside ring theory-->
An algebra is called a '''zero algebra''' if {{nowrap|1=''uv'' = 0}} for all ''u'', ''v'' in the algebra,<ref>{{cite book |first=João B. |last=Prolla |title=Approximation of Vector Valued Functions |chapter=Lemma 4.10 |chapter-url=https://books.google.com/books?id=utTS4nTd-IsC |date=2011 |publisher=Elsevier |isbn=978-0-08-087136-3 |orig-year=1977 |page=65}}</ref> not to be confused with the algebra with one element.  It is inherently non-unital (except in the case of only one element), associative and commutative.

A '''unital zero algebra''' is the [[direct sum]] {{tmath|K\oplus V}} of a field  {{tmath|K}} and a {{tmath|K}}-vector space {{tmath|V}}, that is equipped by the only multiplication that is zero on the vector space (or module), and makes it an unital algebra.

More precisely, every element of the algebra may be uniquely written as {{tmath|k+v}} with {{tmath|k\in K}} and {{tmath|v\in V}}, and the product is the only [[bilinear operation]] such that {{tmath|1=vw=0}} for every {{tmath|v}} and {{tmath|w}} in {{tmath|V}}. So, if {{tmath|k_1, k_2\in K}} and {{tmath|v_1,v_2\in V}}, one has
<math display = block>(k_1+v_1)(k_2+v_2)=k_1k_2 +(k_1v_2+k_2v_1).</math>

A classical example of unital zero algebra is the algebra of [[dual number]]s, the unital zero '''R'''-algebra built from a one dimensional real vector space.

This definition extends verbatim to the definition of a ''unital zero algebra'' over a [[commutative ring]], with the replacement of "field" and "vector space" with "commutative ring" and "[[module (mathematics)|module]]".  

Unital zero algebras allow the unification of the theory of submodules of a given module and the theory of ideals of a unital algebra. Indeed, the submodules of a module {{tmath|V}} correspond exactly to the ideals of {{tmath|K\oplus V}} that are contained in {{tmath|V}}.

For example, the theory of [[Gröbner basis|Gröbner bases]] was introduced by [[Bruno Buchberger]] for [[ideal (ring theory)|ideals]] in a polynomial ring {{nowrap|1=''R'' = ''K''[''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>]}} over a field. The construction of the unital zero algebra over a free ''R''-module allows extending this theory as a Gröbner basis theory for submodules of a free module. This extension allows, for computing a Gröbner basis of a submodule, to use, without any modification, any algorithm and any software for computing Gröbner bases of ideals.

Similarly, unital zero algebras allow to deduce straightforwardly the [[Lasker–Noether theorem]] for modules (over a commutative ring) from the original Lasker–Noether theorem for ideals.

=== Associative algebra ===

{{main|Associative algebra}}
Examples of associative algebras include
* the algebra of all ''n''-by-''n'' [[matrix (mathematics)|matrices]] over a field (or commutative ring) ''K''. Here the multiplication is ordinary [[matrix multiplication]].
* [[Group ring|group algebra]]s, where a [[group (mathematics)|group]] serves as a basis of the vector space and algebra multiplication extends group multiplication.
* the commutative algebra ''K''[''x''] of all [[polynomial]]s over ''K'' (see [[polynomial ring]]).
* algebras of [[function (mathematics)|function]]s, such as the '''R'''-algebra of all real-valued [[continuous function|continuous]] functions defined on the [[interval (mathematics)|interval]] [0,1], or the '''C'''-algebra of all [[holomorphic function]]s defined on some fixed open set in the [[complex plane]]. These are also commutative.
* [[Incidence algebra]]s are built on certain [[partially ordered set]]s.
* algebras of [[linear operator]]s, for example on a [[Hilbert space]]. Here the algebra multiplication is given by the [[functional composition|composition]] of operators. These algebras also carry a [[topological space|topology]]; many of them are defined on an underlying [[Banach space]], which turns them into [[Banach algebra]]s. If an involution is given as well, we obtain [[B*-algebra]]s and [[C*-algebra]]s. These are studied in [[functional analysis]].

=== Non-associative algebra ===

{{main|Non-associative algebra}}

A ''non-associative algebra''<ref name=Schafer>{{cite book |first=Richard D. |last=Schafer|author-link=Richard D. Schafer|title=An Introduction to Nonassociative Algebras |year=1996 |publisher=Courier Corporation |isbn=0-486-68813-5 |url=http://www.gutenberg.org/ebooks/25156}}</ref> (or ''distributive algebra'') over a field ''K'' is a ''K''-vector space ''A'' equipped with a ''K''-[[bilinear map]] <math>A \times A \rightarrow A</math>. The usage of "non-associative" here is meant to convey that associativity is not assumed, but it does not mean it is prohibited – that is, it means "not necessarily associative".

Examples detailed in the main article include:
* [[Euclidean space]] '''R'''<sup>3</sup> with multiplication given by the [[vector cross product]]
* [[Octonion]]s
* [[Lie algebra]]s
* [[Jordan algebra]]s 
* [[Alternative algebra]]s
* [[Flexible algebra]]s
* [[Power-associative algebra]]s