= Ore extension
{c}
{title2=$A[t;\sigma,\delta],\quad ta=\sigma(a)t+\delta(a)$}
{wiki}
An Ore extension uses a unital endomorphism $\sigma$ of $A$ and a $\sigma$-derivation $\delta$, satisfying $\delta(ab)=\sigma(a)\delta(b)+\delta(a)b$. Its elements have unique expressions $\sum_i a_it^i$ and multiplication is determined by the displayed relation. Applying the relation to $t(ab)$ shows why the endomorphism and <derivation> conditions are exactly the compatibility conditions for <associativity>. A differential Ore extension has $\sigma=\operatorname{id}$. Adjoining ordinary partial derivatives successively to a <polynomial ring> gives the ordered-monomial construction of a <Weyl algebra>.
Back to article page
