Take $L$ a $k$-Lie algebra (let $k$ be a commutative ring with unit) and $TL$ its tensor algebra. We have that $$UL=TL/I$$ is the universal enveloping algebra of $L$, where $I$ is the two sided ideal of $TL$ generated by elements of type $$x\otimes y-y\otimes x- [x,y] . $$
First question: Do we need that $I$ is two sided?
Second question Is it correct to say that $UL$ is an associative unitary $k$-algebra because $TL$ is ?
Best Answer
Question: "First question: Do we need that I is two sided? Second question Is it correct to say that UL is an associative unitary algebras because TL is ?"
Answer: First question: $T(L)$ is an associative unital $k$-algebra. For a quotient $U(L)$ of $T(L)$ to be an associative unital $k$-algebra we must take the quotient of $T(L)$ by a 2-sided ideal. Second question: Since $T(L)$ is an associative unital $k$-algebra and since $I$ is a 2-sided ideal, it is correct to say that "$U(L)$ is an associative unital $k$-algebra".
Note: We may let $k$ be any commutative unital ring.
Note: If $x,y\in T(L), u,v\in I$ we want to define a product of classes $\overline{x}\overline{y} \in U(L)$. For this to be well defined, the class of $xy$ should equal the class of
$$(x+u)(y+v)=xy+xv+uy+uv$$
and since $I$ is 2-sided it follows $xv+uy+uv\in I$ and hence the class of $xy$ equals the class of $(x+u)(y+v)$. It follows the multiplication is well defined.