We have an answer to "Let k be a field and R be..."

College
Algebra
Abstract algebra

socorraob

Answered question

2022-01-13

Let k be a field and R be the exterior ring over

k^{d}

, that is, k-algebra genarated by elements

x_{1}, \dots . x_{d},

where

x_{i} x_{j} = - x_{j} x_{i}

?
Is R Artinian?

Answer & Explanation

Bubich13

Beginner2022-01-14Added 36 answers

Step 1
Actually, I am not sure the question is true as stated if
$1 = - 1$ in $k!$
If 2 is invertible, then you get $x_{i}^{2} = 0$ for all i. As suggested in the comments and Mariano's answer, the key point is R has finite dimension as a k-vector space. Since the monomials span R, you want a lot of them to vanish. For example:
$x_{1} x_{2} x_{1} = - x_{1} x_{1} x_{2} = - x_{1}^{2} x_{2} = 0$

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