Given an Associative algebra $A$ what is the definitions of its generators? That is if $A$ an Associative algebra what is the meaning of saying that $A$ is generated by a set $B$.
It would appreciate some reference and examples too.
2026-05-15 09:41:57.1778838117
Generator of a algebra
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Short, intuitive explanation: if you take the elements of $B\subseteq A$, and use any and all available algebra operations on those elements (finitely many times), can you get any element of $A$?
This is how "generated by" usually works in algebra. A group $G$ is generated by $H\subseteq G$ if you can take the elements of $H$, and use all available group operations (inversion and multiplication, finitely many times), and you can get any element of $G$ this way.
Given a ring $R$, an $R$-module $M$ is generated by $N\subseteq M$ if you by using the elements of $N$, along with any module operation (module addition, negation, and scalar multiplication by any element of $R$, finitely many times), can make any element of $M$.
Some quick examples: the group $\Bbb Z[x]$ is generated by $\{1, x, x^2, \ldots\}$, as is the $\Bbb Z$-module $\Bbb Z[x]$. On the other hand, the $\Bbb Z[x]$-module $\Bbb Z[x]$ is generated by $\{1\}$, and the $\Bbb Z$-algebra $\Bbb Z[x]$ is generated by $\{1, x\}$.