2024/08/27 by Zhang, Xiaolei
#Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.2408.14781
It is well-known that a ring is Noetherian if and only if every ascending chain of ideals is stationary, and an integral domain is a PID if and only if every countably generated ideal is principal. We respectively investigate the similar results on S-Noetherian rings and S-∗w-PIDs, where S is a multiplicative subset and ∗ is a star operation. In particular, we gave negative answers to the open questions proposed by Hamed and Hizem \citehh16, Kim and Lim \citekl18, and Lim \citel18 in terms of valuation domains, respectively.