Page "Product of rings" Paragraph 13

If A < sub > i </ sub > in R < sub > i </ sub > is an ideal for each i in I, then A = Π < sub > i in I </ sub > A < sub > i </ sub > is an ideal of R. If I is finite, then the converse is true, i. e. every ideal of R is of this form.

However, if I is infinite and the rings R < sub > i </ sub > are non-zero, then the converse is false ; the set of elements with all but finitely many nonzero coordinates forms an ideal which is not a direct product of ideals of the R < sub > i </ sub >.

The ideal A is a prime ideal in R if all but one of the A < sub > i </ sub > are equal to R < sub > i </ sub > and the remaining A < sub > i </ sub > is a prime ideal in R < sub > i </ sub >.

However, the converse is not true when I is infinite.

For example, the direct sum of the R < sub > i </ sub > form an ideal not contained in any such A, but the axiom of choice gives that it is contained in some maximal ideal which is a fortiori prime.