Editing Cayley–Hamilton theorem (section)

==Abstraction and generalizations==

The above proofs show that the Cayley–Hamilton theorem holds for matrices with entries in any commutative ring {{math|''R''}}, and that {{math|1=''p''(''φ'') = 0}} will hold whenever {{math|''φ''}} is an endomorphism of an {{math|''R''}}-module generated by elements {{math|''e''<sub>1</sub>,...,''e''<sub>''n''</sub>}} that satisfies

<math display="block">\varphi(e_j)=\sum a_{ij}e_i, \qquad j =1, \ldots, n.</math>

This more general version of the theorem is the source of the celebrated [[Nakayama lemma]] in [[commutative algebra]] and [[algebraic geometry]].

The Cayley-Hamilton theorem also holds for matrices over the [[quaternions]], a [[noncommutative ring]].<ref>{{harvnb|Zhang|1997}}</ref><ref group=nb>Due to the non-commutative nature of the multiplication operation for quaternions and related constructions, care needs to be taken with definitions, most notably in this context, for the determinant. The theorem holds as well for the slightly less well-behaved [[split-quaternion]]s, see {{harvtxt|Alagös|Oral|Yüce|2012}}. The rings of quaternions and split-quaternions can both be represented by certain {{math|2&thinsp;×&thinsp;2}} complex matrices. (When restricted to unit norm, these are the [[group (mathematics)|groups]] {{math|SU(2)}} and {{math|SU(1,1)}} respectively.) Therefore it is not surprising that the theorem holds.<br>There is no such matrix representation for the [[octonion]]s, since the multiplication operation is not [[associative]] in this case. However, a modified Cayley–Hamilton theorem still holds for the octonions, see {{harvtxt|Tian|2000}}.</ref>