ACCR ring of the forms \(\mathcal {A}[X]\) and \(\mathcal {A}X\)
Recall that a module \(M\) over a commutative ring \(A\) satisfies ACCR (the ascending chain of residuals) if for every submodule \(N\) of \(M\) and for every finitely generated ideal \(B\) of \(A\), the sequence \((N:B)\subseteq( N:B^2)\subseteq( N:B^3)\dots\) stabilizes. Let \(\mathcal A = (A_ n)_{n\ge0}\) be an increasing sequence of commutative rings. The ring \(\mathcal AX\) is defined as the subring of \(AX\) of power series \(\sum_{n=0}^{\infty}a_nX^n\) such that \(a_n\in A_n\) for all \(n\ge0\). The ring \(\mathcal A[X]\) is equal to \(\mathcal AX\cap A[X]\). The authors prove that if \(A_n\) is a finitely generated free \(A_0\)-module for all \(n\ge1\), then \(\mathcal A[X]\) satisfies ACCR iff \(AX\) satisfies ACCR iff \(A_0\) is Noetherian and the sequence \(A_n\) is stationary. In particular, if a ring \(B\) containing \(A\), is a finitely generated free \(A\)-module, then \(A+XB[X]]\) satisfies ACCR iff \(A+XX\) satisfies ACCR iff \(A\) is Noetherian. Using ring amalgamation, the authors obtain another example of a ring \(\mathcal A[X]\) satisfying ACCR.
scientific article; zbMATH DE number 6983427
scientific article