Commutative Rings Whose Principal Ideals Have Unique Generators

Torsion theories over commutative rings

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Let R be a commutative ring with Z*(R) as the set of non-zero zero divisors. The zero divisor graph of R, denoted by Γ(R), is the graph whose vertex set is Z*(R), where two distinct vertices x and y are adjacent if and only if xy = 0. In this paper, we investigate the metric dimension dim(Γ(R)) and upper dimension dim+(Γ(R)) of zero divisor graphs of commutative rings. For zero divisor graphs Γ(R) associated to finite commutative rings R with unity 1 ≠ 0, we conjecture that dim+(Γ(R)) = dim(Γ(R)), with one exception that R ≅ Π 𝕑 2 n {\rm{R}} \cong \Pi {\rm\mathbb{Z}}_2^{\rm{n}} , n ≥ 4. We prove that this conjecture is true for several classes of rings. We also provide combinatorial formulae for computing the metric and upper dimension of zero divisor graphs of certain classes of commutative rings besides giving bounds for the upper dimension of zero divisor graphs of rings.

INTRODUCTION In the broadest sense, this is a study of commutative rings which satisfy the (finitely) pseudo-Froben[us (or (F)PF) condition: All (finitely generated) faithful modules generate the category mod-R of all R-modules. These rings include: Pr[[fer rings, almost maximal valuation rings, self-injective rings, e.g. , quasi-Frobenius (QF) and pseudo-Frobenius (PF) rings, and finite products of these. (In fact, any product of commutative FPF rings is FPF [34]; hence, any product of commutative PF rings is FPF (cf. §9).) If R is FPF, so is its (classical)ring of quotients Qc~(R) and its maximal quotient ring Qmax(R). All known FPF rings are (classically) quotientinjective in the sense that Qcf is injective. 2 We conjecture that all FIDF rings are quot[ent-injective, and prove this in the three cases: (i) local rings (Proposition 7 and Theorem 9B): (2) Noether[an rings (Theorem ii; Endo' s Theorem [25J~ (3) reduced rings (Proposition 3B and Theorem 4). Moreover, any FPF commutative ring R splits, R = R 1 X R2, where R 1 is semihered[tary, and R 2 has essential nilradical. (If R is semilocal or Noetherian, then R 2 is injective. ) Thus any reduced FPF ring has regular [njective Qc~' and conversely any quotient-injective sem[hered[tary ring is FPF (Theorem 4). A ring iS pre-(l~PF iff all(finitely generated)faithful ideals are generators, and we

Several different generalizations of finite factorization domains (i.e., integral domains where every nonzero nonunit has only finitely many divisors up to associates) have been defined for commutative rings with zero divisors. We study these notions in the context of commutative monoid rings with zero divisors, utilizing semigroup theory to simultaneously generalize and extend many past results about “finite factorization” properties in commutative polynomial rings. Along the way, we expand upon the general theory of factorization in commutative rings with zero divisors, providing new characterizations and results about several kinds of “finite factorization rings.”

We continue the study of intersection algebras [Formula: see text] of two ideals [Formula: see text] in a commutative Noetherian ring [Formula: see text]. In particular, we exploit the semigroup ring and toric structures in order to calculate various invariants of the intersection algebra when [Formula: see text] is a polynomial ring over a field and [Formula: see text] are principal monomial ideals. Specifically, we calculate the [Formula: see text]-signature, divisor class group, and Hilbert–Samuel and Hilbert–Kunz multiplicities, sometimes restricting to certain cases in order to obtain explicit formulæ. This provides a new class of rings where formulæ for the [Formula: see text]-signature and Hilbert–Kunz multiplicity, dependent on families of parameters, are provided.

Let R be a commutative ring with identity. A short exact sequence of R-modules is said to be neat (respectively, -pure) if the sequence (respectively, ) is exact for every simple R-module S. The characterization of the integral domains over which neatness and -purity coincide has been given by László Fuchs: they are the integral domains where every maximal ideal is projective (and so also finitely generated). We show that for a commutative ring R, every maximal ideal of R is finitely generated and projective if and only if R has projective socle and neatness and -purity coincide. For a commutative ring R, we prove that neatness and -purity coincide if and only if every maximal ideal of R is finitely generated and the unique maximal ideal PP of the local ring RP is a principal ideal for every maximal ideal P of R. This result is proved firstly over commutative local rings and then using localization over any commutative ring. The Auslander-Bridger transpose of simple modules is used in proving these equivalences because it is an essential tool for the passage between proper classes of short exact sequences of modules that are projectively generated and these that are flatly generated by a set of finitely presented modules. For a commutative ring R, we prove that neatness and -purity coincide if and only if every simple R-module S is finitely presented and an Auslander-Bridger transpose of S is projectively equivalent to S.

BL-algebras are algebraic structures corresponding to Hajek’s basic fuzzy logic. The aim of this paper is to analyze the structure of BL-algebras using commutative rings. Due to computational considerations, we are interested in the finite case. We present new ways to generate finite BL-algebras using commutative rings and provide summarizing statistics. Furthermore, we investigated BL-rings, i.e., commutative rings whose the lattice of ideals can be equipped with a structure of BL-algebra. A new characterization for these rings and their connections to other classes of rings is established. Furthermore, we give examples of finite BL-rings for which the lattice of ideals is not an MV-algebra and, using these rings, we construct BL-algebras with 2r+1 elements, r≥2, and BL-chains with k elements, k≥4. In addition, we provide an explicit construction of isomorphism classes of BL-algebras of small n size (2≤n≤5).

A commutative ring R is called J-rad clean in case, for any r ∈ R, there is an idempotent e ∈ R such that r−e ∈ U(R) and re ∈ J(R), where U(R) and J(R) denote the set of units and the Jacobson radical of R, respectively. Also, a ring R is called semiregular if R/J(R) is regular in the sense of von Neumann and idempotents lift modulo J(R). We demonstrate that these two concepts are, actually, equivalent and we portray a portion of the properties of this class of rings. In particular, as a direct application, we prove that the commutative group ring RG is J-rad clean if, and only if, R is a commutative J-rad clean ring and G is a torsion abelian group, provided that J(R) is nil.

Let [Formula: see text] be a commutative ring with identity, [Formula: see text] be an indeterminate over [Formula: see text], [Formula: see text] be the polynomial ring over [Formula: see text], [Formula: see text] be the power series ring over [Formula: see text], and [Formula: see text] be a principal prime ideal of [Formula: see text] with [Formula: see text]. It is well known that if [Formula: see text] is an integral domain, then [Formula: see text] is a DVR and [Formula: see text] has infinitely many such principal prime ideals. In this paper, among other things, we show that (i) [Formula: see text] is a field, (ii) [Formula: see text] is a DVR, but (iii) there is a ring [Formula: see text] such that [Formula: see text] has no principal prime ideal. We also study the maximal ideals of [Formula: see text] that are principal.

Let A be a commutative ring with identity and I an ideal of A. A is said to be I-clean if for every element a∈A there is an idempotent e=e 2 ∈A such that a-e is a unit and ae belongs to I. A filter of ideals, say ℱ, of A is Noetherian if for each I∈ℱ there is a finitely generated ideal J∈ℱ such that J⊆I. We characterize I-clean rings for the ideals 0, n(A), J(A), and A, in terms of the frame of multiplicative Noetherian filters of ideals of A, as well as in terms of more classical ring properties.

In this paper, we define and study a valuation dimension for commutative rings. The valuation dimension is a measure of how far a commutative ring deviates from being valuation. It is shown that a ring R with valuation dimension has finite uniform dimension. We prove that a ring R is Noetherian (respectively, Artinian) if and only if the ring R × R has (respectively, finite) valuation dimension if and only if R has (respectively, finite) valuation dimension and all cyclic uniserial modules are Noetherian (respectively, Artinian). We show that the class of all rings of finite valuation dimension strictly lies between the class of Artinian rings and the class of semi-perfect rings.

The main purposes of this paper are to investigate ℤ-injective rings with the representation extension property and its dual, to give a necessary and sufficient condition for a ℤ-injective ring to be an amalgamation base in the class of all rings and to determine structure of ℤ-injective Noetherian rings which are amalgamation bases. Further, in the class of all commutative rings, it is shown that a commutative ring has the representation extension property, if, and only if, it is an amalgamation base.

Unique factorization and small regular local noether lattices

It was shown by C. Wickham [12] that “for a fixed positive integer g, there are finitely many isomorphism classes of finite commutative rings whose zero-divisor graph has genus g.” In this note, we give a short direct proof for this result. Moreover, we show that, if the zero-divisor graph of a commutative ring R has finite genus g, then either g = 0 or R is a finite ring. This immediately generalizes Wickham's theorem to arbitrary (not necessary finite) commutative rings.