On strongly J-Noetherian rings
In this paper, we introduce a class of commutative rings which is a generalization of ZD-rings and rings with Noetherian spectrum. A ring [Formula: see text] is called strongly[Formula: see text]-Noetherian whenever the ring [Formula: see text] is [Formula: see text]-Noetherian for every non-nilpotent [Formula: see text]. We give some characterizations for strongly [Formula: see text]-Noetherian rings and, among the other results, we show that if [Formula: see text] is strongly [Formula: see text]-Noetherian, then [Formula: see text] has Noetherian spectrum, which is a generalization of Theorem 2 in Gilmer and Heinzer [The Laskerian property, power series rings, and Noetherian spectra, Proc. Amer. Math. Soc. 79 (1980) 13–16].
1953 ◽
Vol 49
(3)
◽
pp. 386-396
◽
2013 ◽
Vol 13
(02)
◽
pp. 1350083
◽
Keyword(s):
2009 ◽
Vol 52
(1)
◽
pp. 155-159
◽
2011 ◽
Vol 31
(1)
◽
pp. 331-343
◽
1990 ◽
pp. 271-277
◽
2013 ◽
Vol 24
(2)
◽
Keyword(s):
2019 ◽
Vol 223
(9)
◽
pp. 3980-3988
2011 ◽
Vol 334
(1)
◽
pp. 175-194
◽