scholarly journals Commutative rings whose quotients are Goldie

1975 ◽  
Vol 16 (1) ◽  
pp. 32-33 ◽  
Author(s):  
Victor P. Camillo
Keyword(s):  
Commutative Ring ◽  
Noetherian Ring ◽  
Quotient Ring ◽  
Chain Condition ◽  
Commutative Rings ◽  
The Other ◽  
Direct Sums ◽  
Other Hand ◽  
Commutative Domain ◽  
Ascending Chain Condition

All rings considered here have units. A (non-commutative) ring is right Goldieif it has no infinite direct sums of right ideals and has the ascending chain condition on annihilator right ideals. A right ideal A is an annihilator if it is of the form {a ∈ R/xa = 0 for all x ∈ X}, where X is some subset of R. Naturally, any noetherian ring is Goldie, but so is any commutative domain, so that the converse is not true. On the other hand, since any quotient ring of a noetherian ring is noetherian, it is true that every quotient is Goldie. A reasonable question therefore is the following: must a ring, such that every quotient ring is Goldie, be noetherian? We prove the following theorem:Theorem. A commutative ring is noetherian if and only if every quotient is Goldie.

scholarly journals THE ANNIHILATING-IDEAL GRAPH OF COMMUTATIVE RINGS I

2011 ◽  
Vol 10 (04) ◽  
pp. 727-739 ◽  
Author(s):  
M. BEHBOODI ◽  
Z. RAKEEI
Keyword(s):  
Commutative Ring ◽  
Maximal Ideal ◽  
Connected Graph ◽  
Undirected Graph ◽  
Chain Condition ◽  
Commutative Rings ◽  
Finiteness Conditions ◽  
Descending Chain Condition ◽  
Ascending Chain Condition

Let R be a commutative ring, with 𝔸(R) its set of ideals with nonzero annihilator. In this paper and its sequel, we introduce and investigate the annihilating-ideal graph of R, denoted by 𝔸𝔾(R). It is the (undirected) graph with vertices 𝔸(R)* ≔ 𝔸(R)\{(0)}, and two distinct vertices I and J are adjacent if and only if IJ = (0). First, we study some finiteness conditions of 𝔸𝔾(R). For instance, it is shown that if R is not a domain, then 𝔸𝔾(R) has ascending chain condition (respectively, descending chain condition) on vertices if and only if R is Noetherian (respectively, Artinian). Moreover, the set of vertices of 𝔸𝔾(R) and the set of nonzero proper ideals of R have the same cardinality when R is either an Artinian or a decomposable ring. This yields for a ring R, 𝔸𝔾(R) has n vertices (n ≥ 1) if and only if R has only n nonzero proper ideals. Next, we study the connectivity of 𝔸𝔾(R). It is shown that 𝔸𝔾(R) is a connected graph and diam (𝔸𝔾)(R) ≤ 3 and if 𝔸𝔾(R) contains a cycle, then gr (𝔸𝔾(R)) ≤ 4. Also, rings R for which the graph 𝔸𝔾(R) is complete or star, are characterized, as well as rings R for which every vertex of 𝔸𝔾(R) is a prime (or maximal) ideal. In Part II we shall study the diameter and coloring of annihilating-ideal graphs.

The flat topology on the minimal and maximal prime spectrum of a commutative ring

2019 ◽  
Vol 18 (06) ◽  
pp. 1950110
Author(s):  
Esmaeil Rostami ◽  
Masoumeh Hedayati ◽  
Nosratollah Shajareh Poursalavati
Keyword(s):  
Topological Space ◽  
Commutative Ring ◽  
Chain Condition ◽  
Commutative Rings ◽  
Topological Properties ◽  
Prime Spectrum ◽  
Compact Topological Space ◽  
Algebraic Properties ◽  
Ascending Chain Condition

In this paper, we investigate connections between some algebraic properties of commutative rings and topological properties of their minimal and maximal prime spectrum with respect to the flat topology. We show that for a commutative ring [Formula: see text], the ascending chain condition on principal annihilator ideals of [Formula: see text] holds if and only if [Formula: see text] is a Noetherian topological space as a subspace of [Formula: see text] with respect to the flat topology and we give a characterization for a topological space [Formula: see text] for which [Formula: see text] is a Noetherian topological space as a subspace of [Formula: see text] with respect to the flat topology. Also, we give a characterization for rings whose maximal prime spectrum is a compact topological space with respect to the flat topology. Some other results are obtained too.

scholarly journals Polynomial functions on subsets of non-commutative rings — a link between ringsets and null-ideal sets

2018 ◽  
Vol 20 ◽  
pp. 01003
Author(s):  
Sophie Frisch
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
The Other ◽  
Polynomial Functions ◽  
Other Hand ◽  
Null Ideal

Regarding polynomial functions on a subset S of a non-commutative ring R, that is, functions induced by polynomials in R[x] (whose variable commutes with the coeffcients), we show connections between, on one hand, sets S such that the integer-valued polynomials on S form a ring, and, on the other hand, sets S such that the set of polynomials in R[x] that are zero on S is an ideal of R[x].

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.

Completely Indecomposable Modules

1949 ◽  
Vol 1 (2) ◽  
pp. 125-152 ◽  
Author(s):  
Ernst Snapper
Keyword(s):  
Abelian Group ◽  
Commutative Ring ◽  
Chain Condition ◽  
Indecomposable Module ◽  
Unit Element ◽  
Indecomposable Modules ◽  
Operator Domain ◽  
Unit Operator ◽  
Ascending Chain Condition

The purpose of this paper is to investigate completely indecomposable modules. A completely indecomposable module is an additive abelian group with a ring A as operator domain, where the following four conditions are satisfied.1-1. A is a commutative ring and has a unit element which is unit operator for .1-2. The submodules of satisfy the ascending chain condition. (Submodule will always mean invariant submodule.)

scholarly journals On the semiprimitivity of skew polynomial rings

1993 ◽  
Vol 36 (2) ◽  
pp. 169-178 ◽  
Author(s):  
A. Moussavi
Keyword(s):  
Noetherian Ring ◽  
Chain Condition ◽  
Polynomial Rings ◽  
Skew Polynomial Rings ◽  
Skew Polynomial ◽  
Ascending Chain Condition

Let R be a left Noetherian ring with the ascending chain condition on right annihilators, let α be a ring monomorphism of R and δ an α-derivation of R. We prove that, if R is semiprime or α-prime, then R[X;α, δ] is semiprimitive (and left Goldie), and that J(R[X;α]) equals N(R)[X;α].

On the Nilpotency of Nil Subrings

1970 ◽  
Vol 22 (6) ◽  
pp. 1211-1216 ◽  
Author(s):  
Joe W. Fisher
Keyword(s):  
Left Ideal ◽  
Noetherian Ring ◽  
Short Proof ◽  
Sufficient Conditions ◽  
Chain Condition ◽  
Intermediate Step ◽  
Famous Theorem ◽  
Important Theorem ◽  
Left And Right ◽  
Ascending Chain Condition

A famous theorem of Levitzki states that in a left Noetherian ring each nil left ideal is nilpotent. Lanski [5] has extended Levitzki's theorem by proving that in a left Goldie ring each nil subring is nilpotent. Another important theorem in this area which is due to Herstein and Small [3] states that if a ring satisfies the ascending chain condition on both left and right annihilators, then each nil subring is nilpotent. We give a short proof of a theorem (Theorem 1.6) which yields both Lanski's theorem and Herstein- Small's theorem. We make use of the ascending chain condition on principal left annihilators in order to obtain, at an intermediate step, a theorem (Theorem 1.1) which produces sufficient conditions for a nil subring to be left T-nilpotent.

A Remark on the Krull-Schmidt-Azumaya Theorem

1970 ◽  
Vol 13 (4) ◽  
pp. 501-505 ◽  
Author(s):  
B. L. Osofsky
Keyword(s):  
Dedekind Domain ◽  
The Other ◽  
Group Algebras ◽  
Endomorphism Rings ◽  
Algebraic Integers ◽  
Direct Sums ◽  
Indecomposable Modules ◽  
Direct Sum ◽  
Other Hand

It is well known that if a module M is expressible as a direct sum of modules with local endomorphism rings, then such a decomposition is essentially unique. That is, if M = ⊕i∊IMi = ⊕j∊JNj then there is a bijection f: I → J such that Mi is isomorphic to Nf(i) for all i∊I (see [1]). On the other hand, a nonprincipal ideal in a Dedekind domain provides an example where such a theorem fails in the absence of the local hypothesis. Group algebras of certain groups over rings R of algebraic integers is another such example, where even the rank as R-modules of indecomposable summands of a module is not uniquely determined (see [2]). Both of these examples yield modules which are expressible as direct sums of two indecomposable modules in distinct ways. In this note we construct a family of rings which show that the number of summands in a representation of a module M as a direct sum of indecomposable modules is also not unique unless one has additional hypotheses.

scholarly journals THE ANNIHILATING-IDEAL GRAPH OF COMMUTATIVE RINGS II

2011 ◽  
Vol 10 (04) ◽  
pp. 741-753 ◽  
Author(s):  
M. BEHBOODI ◽  
Z. RAKEEI
Keyword(s):  
Finite Number ◽  
Commutative Ring ◽  
Noetherian Ring ◽  
Undirected Graph ◽  
Commutative Rings ◽  
Complete Characterization ◽  
Zero Divisors ◽  
Reduced Ring

In this paper we continue our study of annihilating-ideal graph of commutative rings, that was introduced in (The annihilating-ideal graph of commutative rings I, to appear in J. Algebra Appl.). Let R be a commutative ring with 𝔸(R) be its set of ideals with nonzero annihilator and Z(R) its set of zero divisors. The annihilating-ideal graph of R is defined as the (undirected) graph 𝔸𝔾(R) that its vertices are 𝔸(R)* = 𝔸(R)\{(0)} in which for every distinct vertices I and J, I — J is an edge if and only if IJ = (0). First, we study the diameter of 𝔸𝔾(R). A complete characterization for the possible diameter is given exclusively in terms of the ideals of R when either R is a Noetherian ring or Z(R) is not an ideal of R. Next, we study coloring of annihilating-ideal graphs. Among other results, we characterize when either χ(𝔸𝔾(R)) ≤ 2 or R is reduced and χ(𝔸𝔾(R)) ≤ ∞. Also it is shown that for each reduced ring R, χ(𝔸𝔾(R)) = cl (𝔸𝔾(R)). Moreover, if χ(𝔸𝔾(R)) is finite, then R has a finite number of minimal primes, and if n is this number, then χ(𝔸𝔾(R)) = cl (𝔸𝔾(R)) = n. Finally, we show that for a Noetherian ring R, cl (𝔸𝔾(R)) is finite if and only if for every ideal I of R with I2 = (0), I has finite number of R-submodules.

A note on the ascending chain condition of ideals

2019 ◽  
Vol 19 (07) ◽  
pp. 2050135 ◽  
Author(s):  
Ibrahim Al-Ayyoub ◽  
Malik Jaradat ◽  
Khaldoun Al-Zoubi
Keyword(s):  
Positive Integer ◽  
Noetherian Ring ◽  
Chain Condition ◽  
Commutative Noetherian Ring ◽  
Ideal Formula ◽  
Regular Ideal ◽  
Reduction Number ◽  
Number Formula ◽  
A Chain ◽  
Ascending Chain Condition

We construct ascending chains of ideals in a commutative Noetherian ring [Formula: see text] that reach arbitrary long sequences of equalities, however the chain does not become stationary at that point. For a regular ideal [Formula: see text] in [Formula: see text], the Ratliff–Rush reduction number [Formula: see text] of [Formula: see text] is the smallest positive integer [Formula: see text] at which the chain [Formula: see text] becomes stationary. We construct ideals [Formula: see text] so that such a chain reaches an arbitrary long sequence of equalities but [Formula: see text] is not being reached yet.

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