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 Incoming links: Truncated polynomial ring

I = (X^{3})_{T}

and

J = (Y^{3})_{R [Y]}

. Part (b)(i) gives a unique representative

a + b X^{2} + c X^{4}

for each element of the quotient ring

T / I

R [Y] / J

likewise has a unique representative

a + bY + c Y^{2}

. Thus the proposed map

Φ

is well-defined and a bijection.

It preserves addition and

1

. To verify multiplication, all terms of polynomial degree at least

5

lie in

I

. Hence

(a + b X^{2} + c X^{4}) (a^{'} + b^{'} X^{2} + c^{'} X^{4}) + I = a a^{'} + (a b^{'} + b a^{'}) X^{2} + (a c^{'} + b b^{'} + c a^{'}) X^{4} + I .

In the other quotient ring,

Y^{3} = 0

(a + bY + c Y^{2}) (a^{'} + b^{'} Y + c^{'} Y^{2}) + J = a a^{'} + (a b^{'} + b a^{'}) Y + (a c^{'} + b b^{'} + c a^{'}) Y^{2} + J .

Therefore

Φ : T / (X^{3})_{T} ≅ R [Y] / (Y^{3}), a + b X^{2} + c X^{4} + I ⟼ a + bY + c Y^{2} + J

is a ring isomorphism. Both sides are examples of a truncated polynomial ring; the generator on the left is the class of

X^{2}

, whose square is nonzero and whose cube vanishes.