Essential sequences and Rees rings
S. McAdam and the author introduced the concept of essential sequences in a Noetherian \(ring\quad R\) (that is, over the ideal \(I=(0)\) of R) which are an excellent analogue of asymptotic sequences in \(R\). But the essential sequences over ideals \(I\neq (0)\) are not a good analogue of asymptotic sequences over \(I\neq (0)\). D. Katz and the author proved that u-essential sequences of \(I\) do play the analogous role to asymptotic sequences over \(I\) and they coincide with essential sequences in \(R\) if \(I=(0)\). It is shown that if \(b_ 1,\dots,b_ s\) is a u-essential sequence over an \(ideal\quad I\) in a Noetherian ring \(R\), then certain permutations \(u, tb_ 1,\dots,tb_ i,b_{i+1},\dots,b_ s\) are a u-essential sequence of \(I{\mathcal R}(R,B_ i)\) and over \(t{\mathcal R}(R,I+B_ i)\) where \(B_ i=(b_ 1,\dots,b_ i)R\) (0\(\leq i\leq s)\) and \({\mathcal R}(R,J)\) is the Rees ring of \(R\) with respect to its ideal \(J\). A number of related results are also given concerning u-essential sequences over \(I\) and over \(ID\) or \((I/b_ k)D\) where \(D\) is the monadic transformation ring \(R[B_ i/b_ k]\) or \(R[(I+B_ i)/b_ k]\); moreover of interest is a containment relation between the essential prime divisors of \(B_ i\) and the u-essential prime divisors of \(I+B_ i\) and the fact that every permutation of \(b_ 1,\dots,b_ s\) is a u- essential sequence over I and is an essential sequence in \(R\).
scientific article; zbMATH DE number 3966250