Comaximal Ideals Research Articles

The annihilators comaximal graph

In this paper, we introduce and study a new kind of graph related to a unitary module [Formula: see text] on a commutative ring [Formula: see text] with identity, namely the annihilators comaximal graph of submodules of [Formula: see text], denoted by [Formula: see text]. The (undirected) graph [Formula: see text] is with vertices of all non-trivial submodules of [Formula: see text] and two vertices [Formula: see text] of [Formula: see text] are adjacent if and only if their annihilators are comaximal ideals of [Formula: see text], i.e. [Formula: see text]. The main purpose of this paper is to investigate the interplay between the graph-theoretic properties of [Formula: see text] and the module-theoretic properties of [Formula: see text]. We study the annihilators comaximal graph [Formula: see text] in terms of the powers of the decomposition of [Formula: see text] to product distinct prime numbers in some special cases.

Commutators of Congruence Subgroups in the Arithmetic Case

In a joint paper of the author with Alexei Stepanov, it was established that for any two comaximal ideals A and B of a commutative ring R, A + B = R, and any n ≥ 3 one has [E(n,R,A),E(n,R,B)] = E(n,R,AB). Alec Mason and Wilson Stothers constructed counterexamples demonstrating that the above equality may fail when A and B are not comaximal, even for such nice rings as ℤ [i]. The present note proves a rather striking result that the above equality and, consequently, also the stronger equality [GL(n,R,A), GL(n,R,B)] = E(n,R,AB) hold whenever R is a Dedekind ring of arithmetic type with infinite multiplicative group. The proof is based on elementary calculations in the spirit of the previous papers by Wilberd van der Kallen, Roozbeh Hazrat, Zuhong Zhang, Alexei Stepanov, and the author, and also on an explicit computation of the multirelative SK1 from the author’s paper of 1982, which, in its turn, relied on very deep arithmetical results by Jean-Pierre Serre and Leonid Vaserstein (as corrected by Armin Leutbecher and Bernhard Liehl). Bibliography: 50 titles.

Commutative Bezout domains in which any nonzero prime ideal is contained in a finite set of maximal ideals

We investigate commutative Bezout domains in which any nonzero prime ideal is contained in a finite set of maximal ideals. In particular, we have described the class of such rings, which are elementary divisor rings. A ring $R$ is called an elementary divisor ring if every matrix over $R$ has a canonical diagonal reduction (we say that a matrix $A$ over $R$ has a canonical diagonal reduction if for the matrix $A$ there exist invertible matrices $P$ and $Q$ of appropriate sizes and a diagonal matrix $D=\mathrm{diag}(\varepsilon_1,\varepsilon_2,\dots,\varepsilon_r,0,\dots,0)$ such that $PAQ=D$ and $R\varepsilon_i\subseteq R\varepsilon_{i+1}$ for every $1\le i\le r-1$). We proved that a commutative Bezout domain $R$ in which any nonze\-ro prime ideal is contained in a finite set of maximal ideals and for any nonzero element $a\in R$ the ideal $aR$ a decomposed into a product $aR = Q_1\ldots Q_n$, where $Q_i$ ($i=1,\ldots, n$) are pairwise comaximal ideals and $\mathrm{rad}\,Q_i\in\mathrm{spec}\, R$, is an elementary divisor ring.

Module Decompositions Using Pairwise Comaximal Ideals

In this article, we show that for a given set of pairwise comaximal ideals {Xi}i∈I in a ring R with unity and any right R-module M with generating set Y and , M = ⊕i∈IC(Xi) if and only if for every y ∈ Y there exists a nonempty finite subset J ⊆ I and positive integers kj such that . We investigate this decomposition for a general class of modules. Our main theorem can be applied to a large class of rings including semilocal rings R with the Jacobson radical of R equal to the prime radical of R, left (or right) perfect rings, piecewise prime rings, and rings with ascending chain condition (ACC) on ideals and satisfying the right AR property on ideals. This decomposition generalizes the decomposition of a torsion abelian group into a direct sum of its p-components. We also develop a torsion theory associated with sets of pairwise comaximal ideals.

On Graphs Related to Comaximal Ideals of a Commutative Ring

We study the co maximal graph Ω(R), the induced subgraph Γ(R) of Ω(R) whose vertex set is R∖(U(R)∪J(R)), and a retract Γr(R) of Γ(R), where R is a commutative ring. For a graph Γ(R) which contains a cycle, we show that the core of Γ(R) is a union of triangles and rectangles, while a vertex in Γ(R) is either an end vertex or a vertex in the core. For a nonlocal ring R, we prove that both the chromatic number and clique number of Γ(R) are identical with the number of maximal ideals of R. A graph Γr(R) is also introduced on the vertex set {Rx∣x∈R∖(U(R)∪J(R))}, and graph properties of Γr(R) are studied.

Standard commutator formula, revisited

Let R be an associative ring with 1; A, B ⊴ R be its ideals; C(n, R, A) be the full congruence subgroup of level A in GL(n, R); and E(n, R, A) be the relative elementary subgroup of level A. We present a very easy proof of the following commutator formula: [E(n, R, A),C(n, R, B)] = [E(n, R, A), E(n, R, B)] for all commutative rings based exclusively on the absolute standard commutator formula and its non-commutative counterparts. This generalizes and strengethens Mason and Stothers’ results, our results, and those of Hazrat and Zhang. For comaximal ideals A + B = R, we show that this commutator is E(n, R, AB + BA).

Graphs associated to co-maximal ideals of commutative rings

Let R be a commutative ring R with 1. In [P.K. Sharma, S.M. Bhatwadekar, A note on graphical representation of rings, J. Algebra 176 (1995) 124–127], Sharma and Bhatwadekar defined a graph on R, Γ ( R ) , with vertices as elements of R, where two distinct vertices a and b are adjacent if and only if a R + b R = R . Let Γ 2 ( R ) be the subgraph of Γ ( R ) induced by non-unit elements. In this paper, we derive several properties of Γ 2 ( R ) . We are able to characterize those rings R for which Γ 2 ( R ) − J ( R ) is a forest and those rings R for which Γ 2 ( R ) − J ( R ) is Eulerian, where J ( R ) is the Jacobson radical of R. We are also able to show that ω ( Γ 2 ( R ) ) = | Max ( R ) | , where ω ( G ) is the clique number of a graph G and Max ( R ) is the set of maximal ideals of R. Moreover, we find all finite rings R such that the genus of Γ 2 ( R ) (resp. Γ ( R ) ) is at most one.

Unique representation domains, II

Given a star operation ∗ of finite type, we call a domain R a ∗ -unique representation domain ( ∗ -URD) if each ∗ -invertible ∗ -ideal of R can be uniquely expressed as a ∗ -product of pairwise ∗ -comaximal ideals with prime radical. When ∗ is the t -operation we call the ∗ -URD simply a URD. Any unique factorization domain is a URD. Generalizing and unifying results due to Zafrullah [M. Zafrullah, On unique representation domains, J. Nat. Sci. Math. 18 (1978) 19–29] and Brewer–Heinzer [J.W. Brewer, W.J. Heinzer, On decomposing ideals into products of comaximal ideals, Comm. Algebra 30 (2002) 5999–6010], we give conditions for a ∗ -ideal to be a unique ∗ -product of pairwise ∗ -comaximal ideals with prime radical and characterize ∗ -URD’s. We show that the class of URD’s includes rings of Krull type, the generalized Krull domains introduced by El Baghdadi and weakly Matlis domains whose t -spectrum is treed. We also study when the property of being a URD extends to some classes of overrings, such as polynomial extensions, rings of fractions and rings obtained by the D + X D S [ X ] construction.

Addition and Subtraction of Ideals

LetAbe a Noetherian ring,ĨandIbe comaximal ideals ofA, andPbe a projectiveA-module. “Addition” refers to being given a surjectionP↠Ĩand producing a surjectionP↠Ĩ∩I. This is useful, for example, whenPis free, to determine how many elements it takes to generate the idealĨ∩I. “Subtraction” refers to being givenP↠Ĩ∩Iand producing someP↠Ĩ. A major use of this is whenĨ=A, to show a projective module has a unimodular element. Here we extend certain addition and subtraction results of R. Sridharan (1995,J. Algebra176,947–958) and S. Mandal and R. Sridharan (1996,J. Math. Kyoto Univ.36,No. 3, 453–470). In both addition and subtraction, we weaken the hypotheses imposed onĨand in a suitable fashion remove the hypothesis thatPmust have trivial determinant. With certain restrictions, we also now allow dimA/I≤1, rather than htI=dimA. WhenAis an affine algebra over a fieldF, Mandal and Sridharan had subtraction results for whenIis the intersection of finitely many maximal ideals whose residue fields are quadratically closed, whenIhas this form (if charF≠2), or whenIis the maximal ideal of anF-rational point. In our generalization, we allowIto be the intersection of finitely many maximal ideals, all of whose residue fields are quadratically closed, except possibly one which instead defines anF-rational point. When charF≠2, we only need to requireIto have this form. In particular, we can subtract an ideal whose radical is the maximal ideal of anF-rational point; this has been used by S. M. Bhatwadekar and R. Sridharan (1998,Invent. Math.133,No. 1, 161–192) to construct a certain local complete intersection ideal which is not a complete intersection ideal.

T-fuzzy prime ideals

The purpose of this paper is to define, study and characterize the T-fuzzy prime ideals. T-fuzzy strongly prime ideals and T-S-fuzzy weakly prime ideals proving the inclusion relationships that are satisfied among them. The T-fuzzy comaximal ideals and fuzzy maximal ideals are also investigated.

The Chinese Remainder Theorem and the Invariant Basis Property

AbstracrtThe Chinese Remainder Theorem states that if I and J are comaximal ideals of a ring R, then A/(I∩J)A is isomorphic to A/IA×A/JA for any left R-module A. In this paper we study the converse; when does A/(I∩J)A and A/IA×A/JA isomorphic imply that I and J are comaximal?

Four-fold torsion theories

In this note 4-fold torsion theories (for categories of modules) are classified by means of orthogonal pairs of comaximal ideals. Among the applications are results of Kurata concerning lengths of n-fold torsion theories and an upper bound for the number of 4-fold torsion theories over a semiperfect ring.