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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.

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.