Editing Algebra over a field (section)

=== 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.