Rings graded by polycyclic-by-finite groups

Chin, William; Quinn, Declan

doi:10.1090/s0002-9939-1988-0920979-3

Cited by 20 publications

(10 citation statements)

References 14 publications

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“…If G is a polycyclic-by-finite group, then graded right noetherian is equivalent with right noetherian [CQ,2.2]; hence strongly graded right noetherian is equivalent with strongly right noetherian. This is not true for all G. For example, a group algebra over a right noetherian ring is always graded right noetherian, but not necessarily right noetherian as an ungraded algebra.…”

Section: Theorem 02 Let G Be a Group Let R Be An Admissible Domainmentioning

confidence: 99%

Generic Flatness for Strongly Noetherian Algebras

1999

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Section: Theorem 02 Let G Be a Group Let R Be An Admissible Domainmentioning

confidence: 99%

Generic Flatness for Strongly Noetherian Algebras

1999

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“…It is not difficult to verify the following fact: if the ring RS" is right Noetherian, then the ring R is also Noetherian (in fact, if It C I2 C .--is an ascending chain of right ideals of R, then I15" C I25" C ... is a chain in RS'). We conclude from this, in view of (3), that R [5"] is right Noetherian. Conversely, if R [5"] is right Noetherian then, in view of (4), RS is right Noetherian.…”

Section: Noetherian Semigroup Ringsmentioning

confidence: 57%

“…We have condition on homogeneous right ideals. It was proved in [3] that for rings graded by polycyclic-by-finite groups, the homogeneous Noether property implies the Noether property. Jespers [15] considered the conditions of the homogeneous Noether property of the rings Rs, where S is a certain submonoid of a polycyclic-by-finite group.…”

Section: Noetherian Semigroup Ringsmentioning

confidence: 99%

Semigroup rings with certain finiteness conditions

Kozhukhov¹

1999

J Math Sci

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“…As above, all sets S^ will be nonempty for 62 < 7 < 6 2 + a n - 2 • If we repeat this reduction n -1 times, we get a set…”

Section: Jmentioning

confidence: 97%

On classical Krull dimension of group-graded rings

Kelarev

1997

Bull. Austral. Math. Soc.

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For any ring R graded by a finite group, we give a bound on the classical Krull dimension of R in terms of the dimension of the initial component R e . It follows that if Re has finite classical Krull dimension, then the same is true of the whole ring R, too.Let G be a finite group with identity e. A ring R is said to be G-graded if R -© R g is a direct sum of additive subgroups R g and R g Rh Q R g h for all g, h € G. g€GThere are many results relating properties of a group-graded ring R = ® R g and its initial component R e , where e is the identity of the group (see [5,7,8] Rings with Krull dimension form an important class and have many nice properties (see [5]). Suppose that the set S = Spec(i?) of prime ideals of R satisfies a.c.c. Define the sets S a inductively. Let SQ be the set of all maximal elements in 5; and for each ordinal a denote by S a the set of all s £ S such that t € 5, t > s implies t E Sp for some /? < a. Then there exists the least ordinal a such that S a = S. This ordinal is called the classical Krull dimension of R. If it is finite, then it is also equal to the right Krull dimension of R defined on the lattice of right ideals of R (see [5, Chapter 6]).Denote by cl-K-dim(-R) the classical Krull dimension of R. For any ordinal a and positive integer n, we introduce ordinals a n , setting c*i = a + 1, a n+i = (a + l)(a n + 1)-We shall use the results on prime ideals due to Cohen and Montgomery [3] and prove the following theorem.

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Rings graded by polycyclic-by-finite groups

Cited by 20 publications

References 14 publications

Generic Flatness for Strongly Noetherian Algebras

Generic Flatness for Strongly Noetherian Algebras

Semigroup rings with certain finiteness conditions

On classical Krull dimension of group-graded rings

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