An element α of a ring is said to be nilpotent if ______________.

In the polynomial ring &mathbb;Q[x], if f(x) = x3 + x2 + x + 1 and g(x) = x3 - x2 + x - 1, then the ideal ⟨f(x), g(x)⟩ generated by f(x) and g(x) is equal to:

Which of the following algebraic structures forms an Abelian group under the usual operation?