The prime avoidance of maximal ideals in commutative rings

1995 ◽  
Vol 23 (3) ◽  
pp. 863-868 ◽  
Author(s):  
V. Erdoĝdu
Keyword(s):  
Commutative Rings ◽  
Maximal Ideals

Annihilating-ideal graphs of commutative rings

2019 ◽  
Vol 13 (07) ◽  
pp. 2050121
Author(s):  
M. Aijaz ◽  
S. Pirzada
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Prime Ideals ◽  
Metric Dimension ◽  
Maximal Ideals ◽  
Vertex Set

Let [Formula: see text] be a commutative ring with unity [Formula: see text]. The annihilating-ideal graph of [Formula: see text], denoted by [Formula: see text], is defined to be the graph with vertex set [Formula: see text] — the set of non-zero annihilating ideals of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] adjacent if and only if [Formula: see text]. Some connections between annihilating-ideal graphs and zero divisor graphs are given. We characterize the prime ideals (or equivalently maximal ideals) of [Formula: see text] in terms of their degrees as vertices of [Formula: see text]. We also obtain the metric dimension of annihilating-ideal graphs of commutative rings.

scholarly journals On 2-absorbing ideals of commutative rings

2007 ◽  
Vol 75 (3) ◽  
pp. 417-429 ◽  
Author(s):  
Ayman Badawi
Keyword(s):  
Prime Ideal ◽  
Maximal Ideal ◽  
Integral Domain ◽  
Commutative Rings ◽  
Dedekind Domain ◽  
Proper Ideal ◽  
Prime Ideals ◽  
Maximal Ideals ◽  
Noetherian Domain ◽  
Valuation Domains

Suppose that R is a commutative ring with 1 ≠ 0. In this paper, we introduce the concept of 2-absorbing ideal which is a generalisation of prime ideal. A nonzero proper ideal I of R is called a 2-absorbing ideal of R if whenever a, b, c ∈ R and abc ∈ I, then ab ∈ I or ac ∈ I or bc ∈ I. It is shown that a nonzero proper ideal I of R is a 2-absorbing ideal if and only if whenever I1I2I3 ⊆ I for some ideals I1,I2,I3 of R, then I1I2 ⊆ I or I2I3 ⊆ I or I1I3 ⊆ I. It is shown that if I is a 2-absorbing ideal of R, then either Rad(I) is a prime ideal of R or Rad(I) = P1 ⋂ P2 where P1,P2 are the only distinct prime ideals of R that are minimal over I. Rings with the property that every nonzero proper ideal is a 2-absorbing ideal are characterised. All 2-absorbing ideals of valuation domains and Prüfer domains are completely described. It is shown that a Noetherian domain R is a Dedekind domain if and only if a 2-absorbing ideal of R is either a maximal ideal of R or M2 for some maximal ideal M of R or M1M2 where M1,M2 are some maximal ideals of R. If RM is Noetherian for each maximal ideal M of R, then it is shown that an integral domain R is an almost Dedekind domain if and only if a 2-absorbing ideal of R is either a maximal ideal of R or M2 for some maximal ideal M of R or M1M2 where M1,M2 are some maximal ideals of R.

CO-MAXIMAL IDEAL GRAPHS OF COMMUTATIVE RINGS

2012 ◽  
Vol 11 (06) ◽  
pp. 1250114 ◽  
Author(s):  
MENG YE ◽  
TONGSUO WU
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Maximal Ideal ◽  
Connected Graph ◽  
Jacobson Radical ◽  
Clique Number ◽  
Commutative Rings ◽  
Maximal Ideals ◽  
Simple Connected Graph

In this paper, a new kind of graph on a commutative ring R with identity, namely the co-maximal ideal graph is defined and studied. We use [Formula: see text] to denote this graph, with its vertices the proper ideals of R which are not contained in the Jacobson radical of R, and two vertices I1 and I2 are adjacent if and only if I1 + I2 = R. We show some properties of this graph. For example, this graph is a simple, connected graph with diameter less than or equal to three, and both the clique number and the chromatic number of the graph are equal to the number of maximal ideals of the ring R.

A new graph associated to a commutative ring

2016 ◽  
Vol 08 (02) ◽  
pp. 1650029 ◽  
Author(s):  
A. Alilou ◽  
J. Amjadi ◽  
S. M. Sheikholeslami
Keyword(s):  
Commutative Ring ◽  
Direct Product ◽  
Connected Graph ◽  
Commutative Rings ◽  
Simple Graph ◽  
Maximal Ideals ◽  
Vertex Set

Let [Formula: see text] be a commutative ring with identity. In this paper, we consider a simple graph associated with [Formula: see text] denoted by [Formula: see text], whose vertex set is the set of all nonzero proper ideals of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent whenever [Formula: see text] or [Formula: see text]. In this paper, we initiate the study of the graph [Formula: see text] and we investigate its properties. In particular, we show that [Formula: see text] is a connected graph with [Formula: see text] unless [Formula: see text] is isomorphic to a direct product of two fields. Moreover, we characterize all commutative rings [Formula: see text] with at least two maximal ideals for which [Formula: see text] are planar.

scholarly journals Invariant maximal ideals of commutative rings

1979 ◽  
Vol 56 (2) ◽  
pp. 472-480
Author(s):  
B.A.F Wehrfritz
Keyword(s):  
Commutative Rings ◽  
Maximal Ideals

scholarly journals A note on universally zero-divisor rings

1991 ◽  
Vol 43 (2) ◽  
pp. 233-239 ◽  
Author(s):  
S. Visweswaran
Keyword(s):  
Zero Divisor ◽  
Chain Condition ◽  
Commutative Rings ◽  
Prime Ideals ◽  
Prüfer Domain ◽  
Integral Extension ◽  
Maximal Ideals ◽  
Unitary Module ◽  
Finite Set ◽  
Descending Chain Condition

In this note we consider commutative rings with identity over which every unitary module is a zero-divisor module. We call such rings Universally Zero-divisor (UZD) rings. We show (1) a Noetherian ring R is a UZD if and only if R is semilocal and the Krull dimension of R is at most one, (2) a Prüfer domain R is a UZD if and only if R has only a finite number of maximal ideals, and (3) if a ring R has Noetherian spectrum and descending chain condition on prime ideals then R is a UZD if and only if Spec (R) is a finite set. The question of ascent and descent of the property of a ring being a UZD with respect to integral extension of rings has also been answered.

A characterization of modules embedding in products of primes and enveloping condition for their class

2015 ◽  
Vol 14 (04) ◽  
pp. 1550051 ◽  
Author(s):  
N. Dehghani ◽  
M. R. Vedadi
Keyword(s):  
Finite Number ◽  
Commutative Rings ◽  
Dedekind Domain ◽  
Noetherian Rings ◽  
Artinian Rings ◽  
Morita Equivalent ◽  
Maximal Ideals ◽  
Weakly Compressible ◽  
Strongly Regular

Modules (cogenerated by nonzero their submodules) having nonzero square homomorphism in nonzero submodules are said (prime) weakly compressible (wc). Such modules are semiprime (i.e. they are cogenerated by their essential submodules). For many rings R, including commutative rings, it is proved that wc modules are isomorphic submodules of products of prime modules, and the converse holds precisely when the class of wc modules is enveloping for mod-R or equivalently, every semiprime module is wc. Semi-Artinian rings R have the latter property and the converse is true when R is strongly regular. Duo Noetherian rings over which wc modules form an enveloping class are shown to have a finite number maximal ideals. If R is Morita equivalent to a Dedekind domain, then the class of wc R-modules is enveloping if and only if R is simple Artinian or J (R) ≠ 0.

Most Commutative Rings Have Maximal Subrings

2012 ◽  
Vol 19 (spec01) ◽  
pp. 1139-1154 ◽  
Author(s):  
A. Azarang ◽  
O. A. S. Karamzadeh
Keyword(s):  
Natural Number ◽  
Direct Product ◽  
Commutative Rings ◽  
Unit Element ◽  
Local Rings ◽  
Maximal Ideals ◽  
Reduced Ring ◽  
Integrally Closed

It is shown that if R is a ring with unit element which is not algebraic over the prime subring of R, then R has a maximal subring. It is shown that whenever R ⊆ T are rings such that there exists a maximal subring V of T, which is integrally closed in T and U(R) ⊈ V, then R has a maximal subring. In particular, it is proved that if R is algebraic over ℤ and there exists a natural number n > 1 with n ∈ U(R), then R has a maximal subring. It is shown that if R is an infinite direct product of certain fields, then the maximal ideals M for which RM (R/M) has maximal subrings are characterized. It is observed that if R is a ring, then either R has a maximal subring or it must be a Hilbert ring. In particular, every reduced ring R with |R|>22ℵ0 or J(R) ≠ 0 has a maximal subring. Finally, the semi-local rings having maximal subrings are fully characterized.

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