Editing Subalgebra (section)

== Subalgebras for algebras over a ring or field ==

A '''subalgebra''' of an [[algebra over a field|algebra over a commutative ring or field]] is a [[vector subspace]] which is closed under the multiplication of vectors. The restriction of the algebra multiplication makes it an algebra over the same ring or field. This notion also applies to most specializations, where the multiplication must satisfy additional properties, e.g. to [[associative algebra]]s or to [[Lie algebra]]s. Only for [[unital algebra]]s is there a stronger notion, of '''unital subalgebra''', for which it is also required that the unit of the subalgebra be the unit of the bigger algebra.

=== Example ===

The 2×2-matrices over the reals '''R''', with [[matrix multiplication]], form a four-dimensional unital algebra M(2,'''R'''). The 2×2-matrices for which all entries are zero, except for the first one on the diagonal, form a subalgebra. It is also unital, but it is not a unital subalgebra.

The [[identity element]] of M(2,'''R''') is the [[identity matrix]] I , so the unital subalgebras contain the line of [[diagonal matrices]] {''x'' I  : ''x'' in '''R'''}. For two-dimensional subalgebras, consider
:<math>E^2 = \begin{pmatrix}a & c \\ b & -a \end{pmatrix}^2 = \begin{pmatrix}a^2+bc & 0 \\ 0 & bc+a^2 \end{pmatrix} = p I \ \ \text{where}\ \ p = a^2 + bc .</math>
When ''p'' = 0, then E is [[nilpotent]] and the subalgebra { ''x'' I + ''y'' E : ''x, y'' in '''R''' } is a copy of the [[dual number]] plane. When ''p'' is negative, take ''q'' = 1/√−''p'', so that (''q'' E)<sup>2</sup> = &minus; I, and subalgebra { ''x'' I + ''y'' (''q''E) : ''x,y'' in '''R''' } is a copy  of the [[complex plane]]. Finally, when ''p'' is positive, take ''q'' = 1/√''p'', so that (''q''E)<sup>2</sup> = I, and subalgebra { ''x'' I + ''y'' (''q''E) : ''x,y'' in '''R''' } is a copy  of the plane of [[split-complex number]]s. By the [[law of trichotomy]], these are the only planar subalgebras of M(2,'''R''').

[[L. E. Dickson]] noted in 1914, the "Equivalence of [[complex quaternion]] and complex matric algebras", meaning M(2,'''C'''), the 2x2 complex matrices.<ref>[[L. E. Dickson]] (1914) ''Linear Algebras'', pages 13,4</ref> But he notes also, "the real quaternion and real matric sub-algebras are not [isomorphic]." The difference is evident as there are the three [[isomorphism class]]es of planar subalgebras of M(2,'''R'''), while real quaternions have only one isomorphism class of planar subalgebras as they are all isomorphic to '''C'''.