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Ideals of $\mathbb Z[x]$

I think the following code should output True instead of False:

R.<x> = ZZ[]
I = R.ideal(1-2*x,2)
I.is_trivial()

Ideals of $\mathbb Z[x]$

I think the following code should output True instead of False because we have $1 = (1-2x)+2x \in I$ :

R.<x> = ZZ[]
I = R.ideal(1-2*x,2)
I.is_trivial()

Ideals of $\mathbb Z[x]$

I think the following code should output True instead of False because we have $1 = (1-2x)+2x \in I$ :I$:

R.<x> = ZZ[]
I = R.ideal(1-2*x,2)
I.is_trivial()

Ideals of $\mathbb Z[x]$

I think the following code should output True instead of False because we have $1 = (1-2x)+2x (1-2x)+2x \in I$:

R.<x> = ZZ[]
I = R.ideal(1-2*x,2)
I.is_trivial()