Topics in graded ring theory

Năstăsescu, C.; Oystaeyen, Freddy Van

doi:10.1007/bfb0067333

Cited by 114 publications

(187 citation statements)

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“…An M G 7<-filt is said to be filt-finitely generated if for all p e Z, FM = 52"i=\E R-Xj for a finite family of x¡ e M, «; g Z, or equivalently if there is a filt-free L that is finitely generated together with a strict surjection n : L -► M. For a filt-finitely generated M we have that G(M) is a finitely generated graded C7(7<)-module but the converse is true only under some additional hypotheses (cf. Proposition IV.3 on p. 294 of [6] and Proposition 1.10 of [1]). If M is filt-finitely generated then we say that FM is good.…”

Section: 4mentioning

confidence: 94%

A New Algebraic Approach to Microlocalization of Filtered Rings

Asensio¹,

Bergh²,

Oystaeyen³

1989

Transactions of the American Mathematical Society

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Abstract.Using the construction of the Rees ring associated to a filtered ring we provide a description of the microlocalization of the filtered ring by using only purely algebraic techniques.The method yields an easy approach towards the study of exactness properties of the microlocalization functor. Every microlocalization at a regular multiplicative Ore set in the associated graded ring can be obtained as the completion of a localization at an Ore set of the filtered ring. IntroductionMicrolocal differential operators appear in the theory of D-modules where they are defined and treated by analytic methods. In search of an algebraic theory of microlocalization, T. Springer [7] introduced algebraic microlocalization of a filtered ring A at a multiplicatively closed subset S in A in the case where the associated graded ring G(A) is a commutative domain. The technique could be extended to microlocalization at a multiplicative subset S of A such that a(S) in G (A) is a left Ore set without further restrictions on the filtered ring A (cf. [8]). The algebraic theory expounded in loc. cit. is not as "algebraic" as most algebraists would like because the methods and results all depend on rather technical calculations with pseudo-norms and completions for these.In this paper we present a purely algebraic approach by passing through the Rees ring R = ®neZFnR, associated to the filtration {FnR,n e Z} on the ring R, and establishing an equivalence of categories between the category Rfilt of filtered left Tc-modules and AT% , the category of X-torsion free graded 7<-modules where X stands for the element of R representing 1 as an element of FXR = Rx . The relations R/(X) = G(R), R/(X -1) = R, and Rx = R[X ,X~ ] allow the interesting interplay on which the results in this paper depend. It is clear that filtered properties of objects in 7<-filt will translate to graded properties in ATX and vice versa; this in its own right provides interesting applications, short elegant proofs of earlier results as well as new perspectives.

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Section: 4mentioning

confidence: 94%

A New Algebraic Approach to Microlocalization of Filtered Rings

Asensio¹,

Bergh²,

Oystaeyen³

1989

Transactions of the American Mathematical Society

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“…For more details concerning graded rings we refer to [4]. Denote by S h (R) the multiplicative semigroup of homogeneous elements of R. For a given ideal I of S h (R) denote byĪ the ideal of R generated by I .…”

Section: Theorem 35 ([2] Theorem) R Is 2-absorbing If and Only If Fmentioning

confidence: 99%

On 2-absorbing commutative semigroups and their applications to rings

Darani

Puczyłowski

2012

Semigroup Forum

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A commutative ring R is called 2-absorbing (Badawi in Bull. Aust. Math. Soc. 75:417-429, 2007) if for arbitrary elements a, b, c ∈ R, abc = 0 if and only if ab = 0 or bc = 0 or ac = 0. In this paper we study this concept in a more general framework of commutative (multiplicative) semigroups with 0. The results obtained apply to many ring theoretic situations and make it possible to describe similarities and differences among some variants of the notion. We pay a particular attention to graded rings. We also show that a conjecture from (Anderson and Badawi in Commun. Algebra 39:1646-1672, 2011) concerning n-absorbing rings holds for rings with torsion-free additive groups.

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“…There 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?)…”

Section: G€gmentioning

confidence: 99%

“…If a prime ideal contains an intersection of a finite number of ideals, then it contains at least one of them. Therefore, for any 8 < e ^ a n and P E S e , there exists Q G Sg, such that Q C. P • For 6 < e < a n , Q G Sg, and P e S e , we shall write if and only if, for a l l / j , 6 < fi < e, we can fix I( £ Sp so that / M C /" whenever 6 ^ fi ^ v ^ e, where Ig -Q and I e = P. We shall show by induction on 7 ^ a n that, for each P £ 5 7 , there exists Q £ Si such that Q < P . The case of 7 = 1 is trivial.…”

mentioning

confidence: 99%

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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Topics in graded ring theory

Cited by 114 publications

References 0 publications

A New Algebraic Approach to Microlocalization of Filtered Rings

A New Algebraic Approach to Microlocalization of Filtered Rings

On 2-absorbing commutative semigroups and their applications to rings

On classical Krull dimension of group-graded rings

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