I get stucked on this problem, hope some one can help me solve this.
Prove that the ring $\mathbb Z[x_{1}, x_{2}, …]/(x_{1}x_{2}, x_{3}x_{4},x_{5}x_{6}, …)$ contains infinitely many minimal prime ideals.
I even can't get the structure and the characteristic of the ring $\mathbb Z[x_{1}, x_{2}, …]/(x_{1}x_{2}, x_{3}x_{4}, …)$.
Thanks.
Best Answer
Write such primes by picking up one variable at a time from the products $x_1x_2$, $x_3x_4$, $x_5x_6$, and so on. (Every prime ideal containing $(x_1x_2,x_3x_4,x_5x_6,\dots)$ must contain one variable from each product.) For example, one of such minimal primes is $P=(x_1,x_4,x_5,\dots)$. It is clear (I hope) that in this way you will find infinitely many minimal primes.