Why require R to be a field to obtain that R[α](2α2) is either an exterior algebra or a polynomial algebra?

Step 1The graded ring Z[α](2α2)with α in degree 1 has 2α2=0 but α2≠0As Andres Mejia points out, you can only deduce that2α2=0⇒α2=0if 2 is invertible in the ring R.By the way, this graded ring has Z in degrees 0 and 1, Z2Z in degree 2.

Expert Answer to "Why require R to be a field to..."

College
Algebra
Abstract algebra

Vandinimo3

Answered question

2022-03-09

Why require R to be a field to obtain that

\frac{R [α]}{(2 α^{2})}

is either an exterior algebra or a polynomial algebra?

Answer & Explanation

sa1yap80

Beginner2022-03-10Added 1 answers

Step 1
The graded ring

\frac{Z [α]}{(2 α^{2})}

with

α

in degree 1 has

2 α^{2} = 0

but

α^{2} \neq 0

As Andres Mejia points out, you can only deduce that

2 α^{2} = 0 \Rightarrow α^{2} = 0

if 2 is invertible in the ring R.
By the way, this graded ring has

Z

in degrees 0 and 1,

\frac{Z}{2 Z}

in degree 2.

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