How is multiplication defined on Grassman ring? I'm reading Kenneth Hoffman's "Linear Algebra", Ed 2. In $\S5.7$ "the Grassman Ring" it briefly mentioned: > The exterior product defines a multiplication product on forms and extend it linearly to $\Lambda(V)$. It distributes over the addition of $\Lambda(V)$ and gives $\Lambda(V)$ the structure of a ring. This ring is the Grassman ring over $V^*$. It is not a commutative ring... But I still wonder how is it defined? Is there an explicit definition on the Grassman ring multiplication I could read?

$\Lambda(V)= \oplus_{k=1}^{\infty}\Lambda^k(V)$ and use multiplication map as $\Lambda^k(V) \times \Lambda^l(V) \rightarrow \Lambda^{k+l}(V)$ so this induces multiplication on $\Lambda(V)$.