In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied...
20 KB (2,773 words) - 10:09, 18 February 2024
is a Noetherian ring. More generally, a scheme is locally Noetherian if it is covered by spectra of Noetherian rings. Thus, a scheme is Noetherian if and...
10 KB (1,306 words) - 18:59, 18 August 2024
length. Noetherian objects are named after Emmy Noether, who was the first to study the ascending and descending chain conditions for rings. Specifically:...
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commutative rings, similar to the role of the finite-dimensional vector spaces in linear algebra. In particular, Noetherian rings (see also ยงย Noetherian rings, below)...
41 KB (5,655 words) - 15:25, 12 December 2023
Artinian ring is Artinian. Since an Artinian ring is also a Noetherian ring, and finitely-generated modules over a Noetherian ring are Noetherian, it is...
8 KB (1,218 words) - 22:28, 4 March 2023
Commutative algebra (redirect from Commutative ring theory)
commutative ring R is Noetherian, the same is true for every polynomial ring over it, and for every quotient ring, localization, or completion of the ring. The...
17 KB (2,020 words) - 15:41, 6 May 2024
Noetherian module. Any module that is finite as a set is Noetherian. Any finitely generated right module over a right Noetherian ring is a Noetherian...
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Krull dimension (redirect from Height (ring theory))
finite even for a Noetherian ring. More generally the Krull dimension can be defined for modules over possibly non-commutative rings as the deviation of...
11 KB (1,745 words) - 22:15, 10 July 2024
full matrix ring M n ( R ) {\displaystyle M_{n}(R)} over a left Artinian (resp. left Noetherian) ring R is left Artinian (resp. left Noetherian). The following...
8 KB (1,252 words) - 22:20, 28 October 2023
Hilbert's basis theorem (category Theorems in ring theory)
ideals have this property are called Noetherian rings. Every field, and the ring of integers are Noetherian rings. So, the theorem can be generalized and...
11 KB (1,845 words) - 13:43, 1 September 2024