Editing Algebra over a field (section)

== Basic concepts ==

=== Algebra homomorphisms ===
Given {{math|''K''}}-algebras {{math|''A''}} and {{math|''B''}}, a [[homomorphism]] of {{math|''K''}}-algebras or {{math|''K''}}-'''algebra homomorphism''' is a {{math|''K''}}-[[linear map]] {{math|''f'': ''A'' → ''B''}} such that {{math|1=''f''(''xy'') = ''f''(''x'') ''f''(''y'')}} for all {{math|''x'', ''y''}} in {{math|''A''}}. If {{math|''A''}} and {{math|''B''}} are unital, then a homomorphism satisfying {{math|1=''f''(1{{sub|''A''}}) = 1{{sub|''B''}}}} is said to be a unital homomorphism. The space of all {{math|''K''}}-algebra homomorphisms between {{math|''A''}} and {{math|''B''}} is frequently written as
:<math>\mathbf{Hom}_{K\text{-alg}} (A,B).</math>
A {{math|''K''}}-algebra [[isomorphism]] is a [[bijective]] {{math|''K''}}-algebra homomorphism.
=== Subalgebras and ideals ===
{{main|Substructure (mathematics)}}
A '''subalgebra''' of an algebra over a field ''K'' is a [[linear subspace]] that has the property that the product of any two of its elements is again in the subspace. In other words, a subalgebra of an algebra is a non-empty subset of elements that is closed under addition, multiplication, and scalar 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.

A ''left ideal'' of a ''K''-algebra is a linear subspace that has the property that any element of the subspace multiplied on the left by any element of the algebra produces an element of the subspace. In symbols, we say that a subset ''L'' of a ''K''-algebra ''A'' is a left ideal if for every ''x'' and ''y'' in ''L'', ''z'' in ''A'' and ''c'' in ''K'', we have the following three statements.
# ''x'' + ''y'' is in ''L'' (''L'' is closed under addition),
# ''cx'' is in ''L'' (''L'' is closed under scalar multiplication),
# ''z'' · ''x'' is in ''L'' (''L'' is closed under left multiplication by arbitrary elements).

If (3) were replaced with ''x'' · ''z'' is in ''L'', then this would define a ''right ideal''. A ''two-sided ideal'' is a subset that is both a left and a right ideal. The term ''ideal'' on its own is usually taken to mean a two-sided ideal. Of course when the algebra is commutative, then all of these notions of ideal are equivalent. Conditions (1) and (2) together are equivalent to ''L'' being a linear subspace of ''A''. It follows from condition (3) that every left or right ideal is a subalgebra.

This definition is different from the definition of an [[ideal (ring theory)|ideal of a ring]], in that here we require the condition (2). Of course if the algebra is unital, then condition (3) implies condition (2).

=== Extension of scalars ===
{{main|Extension of scalars}}
If we have a [[field extension]] ''F''/''K'', which is to say a bigger field ''F'' that contains ''K'', then there is a natural way to construct an algebra over ''F'' from any algebra over ''K''. It is the same construction one uses to make a vector space over a bigger field, namely the tensor product <math> V_F:=V \otimes_K F </math>. So if ''A'' is an algebra over ''K'', then <math>A_F</math> is an algebra over ''F''.