What are the two properties that define a prime ideal in a commutative ring?

The three prime ideals are generated by the integer 3 and the distinct irreducible factors of the polynomial modulo 3. Specifically, $P_1 = (3, x - 1)R$, $P_2 = (3, x + 1)R$, and $P_3 = (3, x^2 + 2x + 2)R$.

Conditions: The ring R is $\mathbb{Z}[x]/(f)$.; The ideals contain $I = 3R$.; The generators correspond to the irreducible factors of $\bar{f}$ in $\mathbb{Z}_3[x]$.

To lift prime ideals from a quotient ring $R/I$ back to the original ring $R$, you take the generators of the prime ideals in $R/I$ and combine them with the generators of the ideal $I$. In this specific problem, the prime ideals in $R$ that contain $I=3R$ are generated by 3 and each of the irreducible factors of the simplified polynomial $\bar{f}$.

Conditions: $R$ is the original ring.; $I = 3R$ is an ideal in $R$.; The irreducible factors of $\bar{f}$ have been found in the quotient ring.

In the context of a polynomial ring over a field modulo a polynomial, the prime ideals of the resulting quotient ring correspond to the ideals generated by the irreducible factors of the modulus polynomial. Specifically, the prime ideals of $\mathbb{Z}_3[x]/\langle \bar{f} \rangle$ are generated by the irreducible factors of $\bar{f}$.

Conditions: The base ring is a polynomial ring over a field.; The modulus polynomial is factored into irreducibles.