Final answer:
The ring Z[x1, x2, x3, ...] / (x1, x2, x3, x4, x5, x6,...) contains infinitely many minimal prime ideals.
Step-by-step explanation:
To prove that the ring Z[x1, x2, x3, ...] / (x1, x2, x3, x4, x5, x6,...) contains infinitely many minimal prime ideals, we can consider the residue classes of the ring modulo the ideals generated by the variables xi. These residue classes, denoted by [a], represent elements in the quotient ring. To show the existence of infinitely many minimal prime ideals, we need to demonstrate that each residue class [a] contains a minimal prime ideal.
Consider the residue class [a] with a fixed value of a. We can construct a minimal prime ideal containing [a] by setting the variables xi to specific values that satisfy certain conditions. For example, if we set x1 = a, x2 = 2a, x3 = 3a, and so on, we can show that the ideal generated by (x1, x2, x3, ...) is a minimal prime ideal containing [a]. This can be proven by showing that any other proper ideal containing [a] must also contain the generator (x1, x2, x3, ...).
Since we can choose infinitely many different values of a, we can construct infinitely many minimal prime ideals in the ring Z[x1, x2, x3, ...] / (x1, x2, x3, x4, x5, x6,...). Therefore, the ring contains infinitely many minimal prime ideals.