Non-commutative polynomial

Non-commutative polynomials are a generalization of polynomials such that different variables do not commute.

definition

Be a crowd and the free monoid over . (Then is ) Be a ring. The non-commutative polynomial ring above is defined as ${\displaystyle {\mathcal {X}}}$ ${\displaystyle W({\mathcal {X}})}$ ${\displaystyle {\mathcal {X}}}$ ${\displaystyle W({\mathcal {X}})=\{x_{1}\cdots x_{n}|x_{i}\in {\mathcal {X}},\;n\geq 1\}\cup \{1\}}$ ${\displaystyle R}$ ${\displaystyle R}$

${\displaystyle R\langle {\mathcal {X}}\rangle :=\{\sum _{x\in W({\mathcal {X}})}r_{w}w|r_{w}\in R,\;r_{w}=0\;\operatorname {f{\ddot {u}}r} {\text{ fast alle }}w\}\cong \bigoplus _{w\in W({\mathcal {X}})}R}$

The addition on becomes component-wise, the multiplication as convolution ${\displaystyle R\langle {\mathcal {X}}\rangle }$

${\displaystyle \sum _{w}a_{w}w\cdot \sum _{w}b_{w}w:=\sum _{w}(\sum _{uv=w}a_{u}b_{v})w}$

Are defined.

properties

• For finite sets one writes . ${\displaystyle {\mathcal {X}}=\{X_{1},\ldots ,X_{n}\}}$ ${\displaystyle R\langle X_{1},\ldots ,X_{n}\rangle }$
• ${\displaystyle R\langle X\rangle =R[X]}$ for a variable ${\displaystyle X}$
• ${\displaystyle R\langle {\mathcal {X}}\rangle /[R\langle {\mathcal {X}}\rangle ,R\langle {\mathcal {X}}\rangle ]=R[{\mathcal {X}}]}$

See also

• polynomial
• non-commutative power series