Some Extensions of Rings with Noetherian Spectrum

In this paper, we study rings with Noetherian spectrum, rings with locally Noetherian spectrum and rings with t-locally Noetherian spectrum in terms of the polynomial ring, the Serre’s conjecture ring, the Nagata ring and the t-Nagata ring. In fact, we show that a commutative ring R with identity has Noetherian spectrum if and only if the Serre’s conjecture ring R[X]^U has Noetherian spectrum, if and only if the Nagata ring R[X]^N has Noetherian spectrum. We also prove that an integral domain D has locally Noetherian spectrum if and only if the Nagata ring D[X]^N has locally Noetherian spectrum. Finally, we show that an integral domain D has t-locally Noetherian spectrum if and only if the polynomial ring D[X] has t-locally Noetherian spectrum, if and only if the t-Nagata ring D[X]^Nv has (t-)locally Noetherian spectrum.