CLEAN, ALMOST CLEAN, POTENT COMMUTATIVE RINGS

2007 ◽  
Vol 06 (04) ◽  
pp. 671-685 ◽  
Author(s):  
K. VARADARAJAN
Keyword(s):  
Commutative Ring ◽  
Analogous Result ◽  
Commutative Rings ◽  
Complete Characterization ◽  
Zariski Topology ◽  
Clean Ring ◽  
Regular Rings

We give a complete characterization of the class of commutative rings R possessing the property that Spec(R) is weakly 0-dimensional. They turn out to be the same as strongly π-regular rings. We considerably strengthen the results of K. Samei [13] tying up cleanness of R with the zero dimensionality of Max(R) in the Zariski topology. In the class of rings C(X), W. Wm Mc Govern [6] has characterized potent rings as the ones with X admitting a clopen π-base. We prove the analogous result for any commutative ring in terms of the Zariski topology on Max(R). Mc Govern also introduced the concept of an almost clean ring and proved that C(X) is almost clean if and only if it is clean. We prove a similar result for all Gelfand rings R with J(R) = 0.

scholarly journals The Logical Complexity of Finitely Generated Commutative Rings

2018 ◽  
Vol 2020 (1) ◽  
pp. 112-166 ◽  
Author(s):  
Matthias Aschenbrenner ◽  
Anatole Khélif ◽  
Eudes Naziazeno ◽  
Thomas Scanlon
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Prime Ideals ◽  
Zariski Topology ◽  
Dual Numbers ◽  
Finitely Generated ◽  
Nonstandard Model

AbstractWe characterize those finitely generated commutative rings which are (parametrically) bi-interpretable with arithmetic: a finitely generated commutative ring A is bi-interpretable with $(\mathbb{N},{+},{\times })$ if and only if the space of non-maximal prime ideals of A is nonempty and connected in the Zariski topology and the nilradical of A has a nontrivial annihilator in $\mathbb{Z}$. Notably, by constructing a nontrivial derivation on a nonstandard model of arithmetic we show that the ring of dual numbers over $\mathbb{Z}$ is not bi-interpretable with $\mathbb{N}$.

On *-clean group rings

2014 ◽  
Vol 14 (01) ◽  
pp. 1550004 ◽  
Author(s):  
Yuanlin Li ◽  
M. M. Parmenter ◽  
Pingzhi Yuan
Keyword(s):  
Local Ring ◽  
Prime Order ◽  
Sufficient Conditions ◽  
Complete Characterization ◽  
Group Rings ◽  
Clean Ring ◽  
Commutative Local Ring ◽  
Irreducible Factorization ◽  
Necessary And Sufficient

A ring with involution * is called *-clean if each of its elements is the sum of a unit and a projection. Clearly a *-clean ring is clean. Vaš asked whether there exists a clean ring with involution * that is not *-clean. In a recent paper, Gao, Chen and the first author investigated when a group ring RG with classical involution * is *-clean and obtained necessary and sufficient conditions for RG to be *-clean, where R is a commutative local ring and G is one of C3, C4, S3 and Q8. As a consequence, the authors provided many examples of group rings which are clean, but not *-clean. In this paper, we continue this investigation and we give a complete characterization of when the group algebra 𝔽Cp is *-clean, where 𝔽 is a field and Cp is the cyclic group of prime order p. Our main result is related closely to the irreducible factorization of a pth cyclotomic polynomial over the field 𝔽. Among other results we also obtain a complete characterization of when RCn (3 ≤ n ≤ 6) is *-clean where R is a commutative local ring.

Commuting linear differential expressions

1981 ◽  
Vol 87 (3-4) ◽  
pp. 331-347 ◽  
Author(s):  
M. Giertz ◽  
M. K. Kwong ◽  
A. Zettl
Keyword(s):  
Commutative Ring ◽  
Algebraic Structure ◽  
Complete Characterization ◽  
Linear Differential

SynopsisWe consider the question: When do two ordinary linear differential expressions commute? It turns out that the set of all expressions which commute with a given one form a commutative ring. Here we study the algebraic structure of these rings. As an application a complete characterization of normal differential expressions is obtained.

scholarly journals On the graph of modules over commutative rings II

Filomat ◽  
2018 ◽  
Vol 32 (10) ◽  
pp. 3657-3665
Author(s):  
Habibollah Ansari-Toroghy ◽  
Shokoufeh Habibi ◽  
Masoomeh Hezarjaribi
Keyword(s):  
Commutative Ring ◽  
Bipartite Graph ◽  
Rocky Mountain ◽  
Commutative Rings ◽  
Star Graph ◽  
Zariski Topology ◽  
Empty Subset ◽  
Topology Graph

Let M be a module over a commutative ring R. In this paper, we continue our study about the quasi-Zariski topology-graph G(?*T) which was introduced in (On the graph of modules over commutative rings, Rocky Mountain J. Math. 46(3) (2016), 1-19). For a non-empty subset T of Spec(M), we obtain useful characterizations for those modules M for which G(?*T) is a bipartite graph. Also, we prove that if G(?*T) is a tree, then G(?*T) is a star graph. Moreover, we study coloring of quasi-Zariski topology-graphs and investigate the interplay between ?(G(?+T)) and ?(G(?+T)).

Graded 1-absorbing prime ideals of graded commutative rings

2021 ◽  
Author(s):  
Mohammed Issoual
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Commutative Rings ◽  
Prime Ideals ◽  
Graded Ring

Let [Formula: see text] be a group with identity [Formula: see text] and [Formula: see text] be [Formula: see text]-graded commutative ring with [Formula: see text] In this paper, we introduce and study the graded versions of 1-absorbing prime ideal. We give some properties and characterizations of these ideals in graded ring, and we give a characterization of graded 1-absorbing ideal the idealization [Formula: see text]

scholarly journals Integral points of bounded degree on affine curves

2015 ◽  
Vol 152 (4) ◽  
pp. 754-768 ◽  
Author(s):  
Aaron Levin
Keyword(s):  
Number Field ◽  
Analogous Result ◽  
Complete Characterization ◽  
Integral Points ◽  
Bounded Degree ◽  
Holomorphic Map ◽  
Affine Curves ◽  
Picard’S Theorem ◽  
Picard's Theorem

We generalize Siegel’s theorem on integral points on affine curves to integral points of bounded degree, giving a complete characterization of affine curves with infinitely many integral points of degree $d$ or less over some number field. Generalizing Picard’s theorem, we prove an analogous result characterizing complex affine curves admitting a nonconstant holomorphic map from a degree $d$ (or less) analytic cover of $\mathbb{C}$.

scholarly journals The quasi-Zariski topology-graph on the maximal spectrum of modules over commutative rings

2018 ◽  
Vol 26 (3) ◽  
pp. 41-56
Author(s):  
H. Ansari-Toroghy ◽  
Sh. Habibi
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Zariski Topology ◽  
Topological Properties ◽  
Maximal Spectrum ◽  
Topology Graph

AbstractLet M be a module over a commutative ring and let Max(M) be the collection of all maximal submodules of M. We topologize Max(M) with quasi-Zariski topology, where M is a Max-top module. For a subset T of Max(M), we introduce a new graph $G(\tau_T^{*m})$, called the quasi-Zariski topology-graph on the maximal spectrum of M. It helps us to study algebraic (resp. topological) properties of M (resp. Max(M)) by using the graphs theoretical tools.

scholarly journals Nilpotents and units in skew polynomial rings over commutative rings

1979 ◽  
Vol 28 (4) ◽  
pp. 423-426 ◽  
Author(s):  
M. Rimmer ◽  
K. R. Pearson
Keyword(s):  
Commutative Ring ◽  
Finite Order ◽  
Polynomial Ring ◽  
Commutative Rings ◽  
Polynomial Rings ◽  
Skew Polynomial Ring ◽  
Nilpotent Elements ◽  
Skew Polynomial Rings ◽  
Skew Polynomial

AbstractLet R be a commutative ring with an automorphism ∞ of finite order n. An element f of the skew polynomial ring R[x, α] is nilpotent if and only if all coefficients of fn are nilpotent. (The case n = 1 is the well-known description of the nilpotent elements of the ordinary polynomial ring R[x].) A characterization of the units in R[x, α] is also given.

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.

scholarly journals On the structure of modules defined by subinjectivity

2019 ◽  
Vol 18 (10) ◽  
pp. 1950188
Author(s):  
Ferhat Altinay ◽  
Engı̇n Büyükaşık ◽  
Yılmaz Durg̃un
Keyword(s):  
Commutative Rings ◽  
Complete Characterization ◽  
Noetherian Rings ◽  
Simple Modules ◽  
Dedekind Domains ◽  
Test Module

The aim of this paper is to present new results and generalize some results about indigent modules. The commutative rings whose simple modules are indigent or injective are fully determined. The rings whose cyclic right modules are indigent are shown to be semisimple Artinian. We give a complete characterization of indigent modules over commutative hereditary Noetherian rings. We show that a reduced module is indigent if and only if it is a Whitehead test module for injectivity over commutative hereditary noetherian rings. Furthermore, Dedekind domains are characterized by test modules for injectivity by subinjectivity.

Sign in / Sign up

Export Citation Format

Share Document