An example from Lang's Algebra about primary ideal

Question

On page 421 in Lang's Algebra, the author writes

Let $R$ be a factorial ring with a prime element $t$. Let $A$ be the subring of polynomials $f(X)∈R[X]$ such that $$f(X)=a_0 + a_1X + \dotsb $$ with $a_1$ divisible by $t$. Let $P=(tX,X^2)$. Then $P$ is prime.

My question is: why $P$ is prime?

Answer 1

The claim is wrong: $P$ is not prime in $A$.

The product of elements $t\in A$ and $X^3\in A$ belongs to $P=(tX,X^2)$, but $t\notin P$ and $X^3\notin P$.

Remark. If instead we take $P=(tX,X^2,X^3)$, then $P$ is prime and $P^2$ is not primary.