The ascending chain condition for real ideals (Q1087588)

Let R(S) denote the smallest real ideal containing a set S. In this paper the author imitates the standard proofs from commutative algebra to show that a ring A satisfies the acc for real ideals if and only if every real prime is R(S) for a finite set S (in fact a one-element set) and that A[X] satisfies the acc for real ideals if and only if A[X] does. He shows by counterexample that this is not true for A[[X]].