Let $V$ be a vector space over $\mathbb{Q}$ and suppose $\dim(V)=\infty$ with basis $\{e_1,e_2,\dots\}$. Let $G$ be the exterior algebra of $V$. How to calculate the Jacobson radical $J(G)$?
(A basis for $G$ is $1$ and all "monomials" $e_{i_1}\wedge e_{i_2}\cdots\wedge e_{i_n}$, where $n>0$ and $i_{1}<i_{2}<\cdots<i_{n}$ and multiplication $e_i\wedge e_j=-e_j\wedge e_i$)
I appreciate any hint. Thank you.
It’s not hard to see that anything with a zero constant term is nilpotent . (As you expand powers, the grades of monomials can only increase, until every one contains the square of a basis vector.)
Then $c+x$ is a unit for every nonzeroconstant $c$ and element $x$ with constant $0$, because $x$ is nilpotent and $c$ is a unit commuting with $x$.
So for every element, if it is not a unit, it is contained in $(e_1,e_.2\ldots)$, which is additively closed. Since the nonunits are closed under addition, the ring is a local ring. Therefore the Jacobson radical is exactly that ideal.