Non-commutative power series

Non-commutative power series represent a generalization of the formal power series in such a way that different variables do not commute.

definition

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

${\displaystyle R\langle \langle {\mathcal {X}}\rangle \rangle :=\{\sum _{x\in W({\mathcal {X}})}r_{w}w|r_{w}\in R\}\cong \prod _{w\in W({\mathcal {X}})}R}$

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

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

Are defined.

properties

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

See also

• non-commutative polynomial