Rings with ascending chain conditions on ideals of type \(n\) (Q1911753)

In this paper the author looks at commutative rings having the ``\(n\)-acc property'', which means that every ascending chain of ideals, such that each ideal has at most \(n\) generators eventually stabilizes. Examples are: \(k[x_1, x_2, \dots, x_n, \dots]\) or \(\mathbb{Q} + X \mathbb{C} [X]\) or \(\mathbb{Z} + pX \mathbb{Z} [X]\). All over the paper the author considers ring extensions like \(A \subseteq B\) such that \(B\) has the ``\(n\)-acc property'' and the rings have a common ideal \({\mathcal J}\). Conditions are sought to make the descent work, i.e. going-down \(A\) has the ``\(n\)-acc property''. Such results are proved for rings of the type \(R + {\mathcal J} B\) and \(D + {\mathcal J}\) in section 2. Another example of such results is theorem 3-1: Let \(A \subseteq B\) be rings such that \(B\) has the ``\(n\)-acc property'', and let the rings have a common ideal \({\mathcal J}\). Suppose that \(B\) is of finite type over \(A\) and every \(A/{\mathcal J}\) module of finite type has the ``\(n\)-acc property''. Then \(A\) has the ``\(n\)-acc property''.