A note on commutative Baer rings

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.

For a commutative local ring R, we know that every ideal is a direct sum of local modules if and only if the unique maximal ideal of R is of the form , where and are principal ideal rings and for all γ’s. This motivates us to determine a commutative ring, not necessarily local, in which every (proper) ideal is a direct sum of local modules. We prove that such a ring is exactly a finite direct product of local rings R with the unique maximal ideal in the above mentioned form. Moreover, we characterize more precisely certain commutative rings such as Noetherian rings, reduced rings, Rickart rings, semi-Artinian rings and self-injective rings, in which every (proper) ideal is a direct sum of local modules.

Let R be a commutative ring with 1 ≠ 0 and n a positive integer. In this article, we study two generalizations of a prime ideal. A proper ideal I of R is called an n-absorbing (resp., strongly n-absorbing) ideal if whenever x 1…x n+1 ∈ I for x 1,…, x n+1 ∈ R (resp., I 1…I n+1 ⊆ I for ideals I 1,…, I n+1 of R), then there are n of the x i 's (resp., n of the I i 's) whose product is in I. We investigate n-absorbing and strongly n-absorbing ideals, and we conjecture that these two concepts are equivalent. In particular, we study the stability of n-absorbing ideals with respect to various ring-theoretic constructions and study n-absorbing ideals in several classes of commutative rings. For example, in a Noetherian ring every proper ideal is an n-absorbing ideal for some positive integer n, and in a Prüfer domain, an ideal is an n-absorbing ideal for some positive integer n if and only if it is a product of prime ideals.

The rings of this paper are assumed to have relations of orthogonality defined on them. Such relations are uniquely determined by complete boolean algebras of ideals. Using the Stone space of these boolean algebras, and following J. Dauns and K.H. Hofmann, a sheaf-theoretic representation is obtained for rings with orthogonality relations, and the rings of global sections of these sheaves are characterized. Baer rings, f–rings and commutative semi-prime rings have natural orthogonality relations and among these the Baer rings are isomorphic to their associated rings of global sections. A special type of ideal is singled out in commutative semi-prime rings and following G. Spirason and E. Strzelecki, in an unpublished note, a characterization of a class of such rings is obtained.

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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

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).

We investigate the class of commutative unital rings in which principal ideals have unique generators. We prove that this class forms a finitely axiomatizable, relatively ideal distributive quasivariety, and also that it equals the quasivariety generated by the class of integral domains with trivial unit group.

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.

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.

We investigate how one can construct pseudo-chain rings, a class of commutative rings that includes pseudo-valuation rings, via the method of idealization. Furthermore, we determine the parameters for the adjacency codes of zero-divisor graphs (or compressed zero-divisor graphs) over certain pseudo-chain rings.

A ring R has the (A)-property (resp., strong (A)-property) if every finitely generated ideal of R consisting entirely of zero divisors (resp., every finitely generated ideal of R generated by a finite number of zero-divisors elements of R) has a nonzero annihilator. The class of commutative rings with property (A) is quite large; for example, Noetherian rings, rings whose prime ideals are maximal, the polynomial ring R[x] and rings whose total ring of quotients are von Neumann regular. Let f : A → B be a ring homomorphism and J be an ideal of B. In this paper, we investigate when the (A)-property and strong (A)-property are satisfied by the amalgamation of rings denoted by A ⋈fJ, introduced by D'Anna, Finocchiaro and Fontana in [3]. Our aim is to construct new original classes of (A)-rings that are not strong (A)-rings, (A)-rings that are not Noetherian and (A)-rings whose total ring of quotients are not Von Neumann regular rings.

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.

The notion of a Boolean algebra with operators (BAO) was first defined by Jonsson and Tarski in 1951. Since that time, many varieties of BAOs, including modal algebras, closure algebras, monadic algebras, and of course relation algebras, have been studied. With the exception of relation algebras, these varieties each have one unary operator. This paper investigates a variety of BAOs with one binary operator. Begin with a semigroup S = $(S,\cdot).$ The complex algebra of S, denoted S$\sp+,$ is a Boolean algebra whose underlying set is the power set of S with set union, intersection and complementation as the Boolean algebra operations. The multiplication operation defined on the semigroup induces a normal, associative binary operation, '*', on the complex algebra as follows: for all subsets A and B contained in S, $A\* B=\{a\cdot b:a\in A, b\in B\}.$ Hence, the complex algebra of a semigroup is a BAO with one normal binary associative operator. Let S$\sp+$ be the class of all complex algebras of semigroups and consider the variety generated by the class S$\sp+,$ denoted ${\cal V}$(S$\sp+).$ The tools required to study this variety are developed, including the duality between BAOs and relational structures as it applies to ${\cal V}$(S$\sp+).$ A closure operator is defined which is used to determine homomorphic images of members of ${\cal V}$(S$\sp+).$ Theorems on the subdirectly irreducible and simples algebras in ${\cal V}$(S$\sp+)$ are proved. Next, the structure of ${\cal V}$(S$\sp+)$ is analyzed. The general problem of representing a BAO with one binary, normal, associative operator as a member of ${\cal V}$(S$\sp+)$ is discussed. Several examples and theorems concerning representation are presented. It is shown that the quasivariety generated by S$\sp+$ is strictly contained in ${\cal V}$(S$\sp+).$ Lastly, the structure of the lattice of subvarieties of ${\cal V}$(S$\sp+)$ is investigated. There are precisely two atoms in this lattice; each atom is generated by a two element algebra. Two infinite chains of varieties exist, with the smallest element in each chain a cover of exactly one atom. This leads to a discussion of certain splitting algebras and conjugate varieties. Finally, equations characterizing some important subvarieties are developed.