scholarly journals Three examples concerning the ore condition in Noetherian rings

1980 ◽  
Vol 23 (2) ◽  
pp. 187-192 ◽  
Author(s):  
A. W. Chatters
Keyword(s):  
Noetherian Ring ◽  
Zero Divisor ◽  
Quotient Ring ◽  
Idempotent Element ◽  
Noetherian Rings ◽  
Regular Elements ◽  
Ore Condition ◽  
The Right

A ring R is said to satisfy the right Ore condition with respect to a subset C of R if, given a ∈ R and e ∈ C, there exist b ∈ R and D ∈ C such that ad = cb. It is well known that R has a classical right quotient ring if and only if R satisfies the right Ore condition with respect to C when C is the set of regular elements of R (a regular elemept of R being an element of R which is not a zero-divisor). It is also well known that not every ring has a classical right quotient ring. If we make the non-trivial assumption that R has a classical right quotient ring, it is natural to ask whether this property also holds in certain rings related to R such as the ring Mn(R) of all n by n matrices over R. Some answers to this question are known when extra assumptions are made. For example, it was shown by L. W. Small in (5) that if R has a classical right quotient ring Q which is right Artinian then Mn(Q) is the right quotient ring of Mn(R) and eQe is the right quotient ring of eRe where e is an idempotent element of R. Also it was shown by C. R. Hajarnavis in (3) that if R is a Noetherian ring all of whose ideals satisfy the Artin-Rees property then R has a quotient ring Q and Mn(Q) is the quotient ring of Mn(R).

scholarly journals A note on Noetherian orders in Artinian rings

1979 ◽  
Vol 20 (2) ◽  
pp. 125-128 ◽  
Author(s):  
A. W. Chatters
Keyword(s):  
Noetherian Ring ◽  
Quotient Ring ◽  
Triangular Matrix ◽  
Idempotent Element ◽  
Noetherian Rings ◽  
Central Idempotent ◽  
Matrix Rings ◽  
Regular Elements ◽  
Zero Divisors ◽  
Left And Right

Throughout this note, rings are associative with identity element but are not necessarily commutative. Let R be a left and right Noetherian ring which has an Artinian (classical) quotient ring. It was shown by S. M. Ginn and P. B. Moss [2, Theorem 10] that there is a central idempotent element e of R such that eR is the largest Artinian ideal of R. We shall extend this result, using a different method of proof, to show that the idempotent e is also related to the socle of R/N (where N, throughout, denotes the largest nilpotent ideal of R) and to the intersection of all the principal right (or left) ideals of R generated by regular elements (i.e. by elements which are not zero-divisors). There are many examples of left and right Noetherian rings with Artinian quotient rings, e.g. commutative Noetherian rings in which all the associated primes of zero are minimal together with full or triangular matrix rings over such rings. It was shown by L. W. Small that if R is any left and right Noetherian ring then R has an Artinian quotient ring if and only if the regular elements of R are precisely the elements c of R such that c + N is a regular element of R/N (for further details and examples see [5] and [6]). By the largest Artinian ideal of R we mean the sum of all the Artinian right ideals of R, and it was shown by T. H. Lenagan in [3] that this coincides in any left and right Noetherian ring R with the sum of all the Artinian left ideals of R.

Localization in Non-Commutative Noetherian Rings

1976 ◽  
Vol 28 (3) ◽  
pp. 600-610 ◽  
Author(s):  
Bruno J. Müller
Keyword(s):  
Prime Ideal ◽  
Noetherian Ring ◽  
Jacobson Radical ◽  
Quotient Ring ◽  
Noetherian Rings ◽  
Regular Elements ◽  
Semiprime Ideal ◽  
Ore Condition

To construct a well behaved localization of a noetherian ringRat a semiprime ideal S, it seems necessary to assume that the set(S)of moduloSregular elements satisfies the Ore condition ; and it is convenient to require the Artin Rees property for the Jacobson radical of the quotient ringRsin addition: one calls such 5classical.To determine the classical semiprime ideals is no easy matter; it happens frequently that a prime ideal fails to be classical itself, but is minimal over a suitable classical semiprime ideal.

On the Ring of Quotients at a Prime Ideal of a Right Noetherian Ring

1972 ◽  
Vol 24 (4) ◽  
pp. 703-712 ◽  
Author(s):  
A. G. Heinicke
Keyword(s):  
Prime Ideal ◽  
Noetherian Ring ◽  
Quotient Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Torsion Class ◽  
Ring Of Quotients ◽  
Ore Condition ◽  
The Right ◽  
Necessary And Sufficient

J. Lambek and G. Michler [3] have initiated the study of a ring of quotients RP associated with a two-sided prime ideal P in a right noetherian ring R. The ring RP is the quotient ring (in the sense of [1]) associated with the hereditary torsion class τ consisting of all right R-modules M for which HomR(M, ER(R/P)) = 0, where ER(X) is the injective hull of the R-module X.In the present paper, we shall study further the properties of the ring RP. The main results are Theorems 4.3 and 4.6. Theorem 4.3 gives necessary and sufficient conditions for the torsion class associated with P to have property (T), as well as some properties of RP when these conditions are indeed satisfied, while Theorem 4.6 gives necessary and sufficient conditions for R to satisfy the right Ore condition with respect to (P).

Semicritical Rings and the Quotient Problem

1984 ◽  
Vol 27 (2) ◽  
pp. 160-170
Author(s):  
Karl A. Kosler
Keyword(s):  
Quotient Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Prime Ideals ◽  
Regular Elements ◽  
The Right ◽  
Minimal Prime ◽  
Necessary And Sufficient ◽  
Quotient Rings ◽  
The Relationship

AbstractThe purpose of this paper is to examine the relationship between the quotient problem for right noetherian nonsingular rings and the quotient problem for semicritical rings. It is shown that a right noetherian nonsingular ring R has an artinian classical quotient ring iff certain semicritical factor rings R/Ki, i = 1,…,n, possess artinian classical quotient rings and regular elements in R/Ki lift to regular elements of R for all i. If R is a two sided noetherian nonsingular ring, then the existence of an artinian classical quotient ring is equivalent to each R/Ki possessing an artinian classical quotient ring and the right Krull primes of R consisting of minimal prime ideals. If R is also weakly right ideal invariant, then the former condition is redundant. Necessary and sufficient conditions are found for a nonsingular semicritical ring to have an artinian classical quotient ring.

The Quotient Problem for Noetherian Rings

1981 ◽  
Vol 33 (3) ◽  
pp. 734-748 ◽  
Author(s):  
Bruno J. Müller
Keyword(s):  
Noetherian Ring ◽  
Quotient Ring ◽  
Noetherian Rings ◽  
Associated Primes ◽  
Singular Proposition ◽  
Role Taking ◽  
Composition Factors

Our work was motivated by attempts to find a criterion for the existence of a classical quotient ring, for a noetherian ring, in analogy with the various known criteria for the existence of an artinian classical quotient ring ([9], [10], [13], [2]).We have restricted our attention to Krull symmetric noetherian rings R, and we make heavy use of the fact that all their Krull composition factors are non-singular (Proposition 7). The collection Kprime R of the associated primes of the Krull composition factors of R plays a central role, taking the place of the collection of the associated primes of R.

A note on (-1, 1) ring satisfying a.c.c

2014 ◽  
Vol 3 (2) ◽  
pp. 34
Author(s):  
Jayalakshmi K.
Keyword(s):  
Banach Algebra ◽  
Quotient Ring ◽  
Chain Condition ◽  
Direct Sums ◽  
Regular Elements ◽  
Finite Dimensional ◽  
Complex Banach Algebra ◽  
The Right ◽  
Ascending Chain Condition

Suppose that a semiprime (-1, 1) ring \(R\) is associative, satisfies the ascending chain condition for the right annihilators of the form \(r(w)\), where $w$ belongs to the nucleus \(N(R)\) and \(R\) contains no infinite direct sums of nonzero right ideals. Then the right quotient ring of $R$ relative to the subset \(W = \lbrace w \in N(R) / w \) is regular in \(R\rbrace\) exist and it is semisimple and artinian. Also if \(A\) be a nonassociative complex Banach algebra which satisfies ascending chain condition on left ideals and assume that the center \(Z(A)\) of \(A\) consists of regular elements then \(Z(A)\cong \mathbb{C}\). As a result if \(A\) be a (-1, 1) noetherian complex Banach algebra then \(A\) is finite-dimensional.

scholarly journals Injective homogeneity and the Auslander–Gorenstein property

1995 ◽  
Vol 37 (2) ◽  
pp. 191-204 ◽  
Author(s):  
Zhong Yi
Keyword(s):  
Noetherian Ring ◽  
Projective Dimension ◽  
Global Dimension ◽  
Noetherian Rings ◽  
Connected Component ◽  
Injective Dimension ◽  
The Right

In this paper we refer to [13] and [16] for the basic terminology and properties of Noetherian rings. For example, an FBNring means a fully bounded Noetherian ring [13, p. 132], and a cliqueof a Noetherian ring Rmeans a connected component of the graph of links of R[13, p. 178]. For a ring Rand a right or left R–module Mwe use pr.dim.(M) and inj.dim.(M) to denote its projective dimension and injective dimension respectively. The right global dimension of Ris denoted by r.gl.dim.(R).

m-potent commutators involving skew derivations and multilinear polynomials

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Mohammad Ashraf ◽  
Sajad Ahmad Pary ◽  
Mohd Arif Raza
Keyword(s):  
Prime Ring ◽  
Quotient Ring ◽  
Multilinear Polynomial ◽  
Extended Centroid ◽  
Prime Rings ◽  
Skew Derivation ◽  
Martindale Quotient Ring ◽  
Multilinear Polynomials ◽  
The Right ◽  
The Relationship

AbstractLet {\mathscr{R}} be a prime ring, {\mathscr{Q}_{r}} the right Martindale quotient ring of {\mathscr{R}} and {\mathscr{C}} the extended centroid of {\mathscr{R}}. In this paper, we discuss the relationship between the structure of prime rings and the behavior of skew derivations on multilinear polynomials. More precisely, we investigate the m-potent commutators of skew derivations involving multilinear polynomials, i.e.,\big{(}[\delta(f(x_{1},\ldots,x_{n})),f(x_{1},\ldots,x_{n})]\big{)}^{m}=[% \delta(f(x_{1},\ldots,x_{n})),f(x_{1},\ldots,x_{n})],where {1<m\in\mathbb{Z}^{+}}, {f(x_{1},x_{2},\ldots,x_{n})} is a non-central multilinear polynomial over {\mathscr{C}} and δ is a skew derivation of {\mathscr{R}}.

scholarly journals The Profile of Cognitive Conflict with Intervention in Understanding Algebra of Students at Pangkep State Polytechnic of Agriculture

2021 ◽  
Vol 4 (2) ◽  
Author(s):  
Anita Nita Sari
Keyword(s):  
Research Method ◽  
Zero Divisor ◽  
Cognitive Conflict ◽  
Quadratic Inequality ◽  
New Information ◽  
Determining Set ◽  
Written Test ◽  
The Right ◽  
Descriptive Qualitative

ABSTRACTThe study aims at obtaining information on the profile of cognitive conflict of students by giving intervention on understanding Algebra of students at Pangkep State Polytechnic of Agriculture. The research method employed descriptive qualitative. The study involved the students who experienced cognitive conflict as a sample. The instruments used in collecting the data were written test and interview. Each of the students delivered his or her answer; he or she would be given new information that could trigger cognitive conflict.The result of the study reveal that (1) the students experienced cognitive conflict in determining set of completion on inequality that did not have a zero divisor. Based on students understanding, quadratic inequality that difficult to be factored or the factors were not integers that did not have solutions, (2) the students experienced cognitive conflict in solving equation that had infinite solutions. The students tended to work procedurally without identifying relational elements formed by the expressions. The subjects did not see the objects produce in first step that showed experession on the left was equal to the expression on the right side, (3) the students experienced cognitive conflict in determining set of completion on inequality segment.ABSTRAK Penelitian ini bertujuan untuk memperoleh informasi tentang profil konflik kognitif mahasiswa dengan pemberian intervensi terhadap pemahaman aljabar mahasiswa Politeknik Pertanian Negeri Pangkep. Metode penelitian yang digunakan adalah deskriptif kualitatif. Penelitian ini melibatkan mahasiswa yang mengalami konflik kognitif sebagai sampel. Untuk pengumpulan data, instrumen yang digunakan adalah soal tertulis dan wawancara. Setiap mahasiswa selesai menyampaikan jawaban, akan diberikan informasi baru yang dapat memicu terjadinya konflik kognitif.Hasil penelitian menunjukkan bahwa: (1) Mahasiswa mengalami konflik kognitif dalam menentukan himpunan penyelesaian pada pertidaksamaan  yang tidak memiliki pembuat nol dan faktor-faktornya bukan bilangan real. Menurut pemahaman mahasiswa pertidaksamaan kuadrat yang sulit untuk difaktorkan tidak memiliki solusi (2) Mahasiswa mengalami konflik kognitif dalam menyelesaikan persamaan yang memiliki solusi yang tak berhingga. Mahasiswa cenderung bekerja secara prosedural tanpa mengindentifikasi elemen-elemen relasional yang dibentuk pada ekspresi tersebut. Subjek tidak memandang objek yang dihasilkan pada langkah pertama yang memperlihatkan bahwa ekspresi di ruas kiri sama dengan ekspresi di ruas kanan (3) Mahasiswa mengalami konflik kognitif dalam menentukan himpunan penyelesaian pada pertidaksamaan setelah diintervensi dengan informasi baru dengan menarik akar pada kedua ruas pertidaksamaan.

scholarly journals When Are Graded Rings Graded S-Noetherian Rings

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1532
Author(s):  
Dong Kyu Kim ◽  
Jung Wook Lim
Keyword(s):  
Noetherian Ring ◽  
Semigroup Ring ◽  
Noetherian Rings ◽  
Homogeneous Ideal ◽  
Commutative Monoid ◽  
Graded Rings ◽  
Graded Ring ◽  
Special Case

Let Γ be a commutative monoid, R=⨁α∈ΓRα a Γ-graded ring and S a multiplicative subset of R0. We define R to be a graded S-Noetherian ring if every homogeneous ideal of R is S-finite. In this paper, we characterize when the ring R is a graded S-Noetherian ring. As a special case, we also determine when the semigroup ring is a graded S-Noetherian ring. Finally, we give an example of a graded S-Noetherian ring which is not an S-Noetherian ring.

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