Let k be a field and R be the exterior

socorraob

socorraob

Answered question

2022-01-13

Let k be a field and R be the exterior ring over kd, that is, k-algebra genarated by elements
x1,.xd,
where
xixj=xjxi?
Is R Artinian?

Answer & Explanation

Bubich13

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 xi2=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:
x1x2x1=x1x1x2=x12x2=0

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