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Let R be a commutative ring and M be a unital R-module. Then M is said to be quasi-Artinian if it contains an essential Artinian submodule. M is called Dedekind finite if whenever N is a submodule of M such that M is isomorphic to the module M ⊕N, then N = 0. The ring R is called DQA-ring if any Dedekind finite R-module is quasi-Artinian. In this note we show that a commutative ring R is a DQA-ring if and only if it is an Artinian principal ideal ring.

Let R be a commutative ring with and let = {R|R is a commutative ring and Nil(R) is a divided prime ideal}. If , then R is called a -ring. In this paper, we introduce the concepts of -torsion modules, -flat modules, and -von Neumann regular rings.

We characterise polynomials f with integer coefficients such that a ring with unity R is necessarily commutative if f(x) is central for all x Ɛ R. We also solve the corresponding problem without the assumption that the ring has a unity.

Let R be a commutative ring with 1 6= 0, I a proper ideal of R, and ∼ a multiplicative congruence relation on R. Let R/∼ = { [x]∼ | x ∈ R } be the commutative monoid of ∼-congruence classes under the induced multiplication [x]∼[y]∼ = [xy]∼, and let Z(R/∼) be the set of zero-divisors of R/∼. The ∼-zero-divisor graph of R is the (simple) graph Γ∼(R) with vertices Z(R/∼) \{[0]∼} and with distinct vertices [x]∼ and [y]∼ adjacent if and only if [x]∼[y]∼ = [0]∼. Special cases include the usual zero-divisor graphs Γ(R) and Γ(R/I), the ideal-based zero-divisor graph ΓI (R), and the compressed zero-divisor graphs ΓE(R) and ΓE(R/I). In this paper, we investigate the structure and relationship between the various ∼-zero-divisor graphs.

Let $R$ be an associative ring with identity, $S$ a multiplicative subset of $R$, and $M$ a right $R$-module. Then $M$ is called an $S$-Noetherian module if for each submodule $N$ of $M$, there exist an element $s \in S$ and a finitely generated submodule $F$ of $M$ such that $Ns \subseteq F \subseteq N$, and $R$ is called a right $S$-Noetherian ring if $R_R$ is an $S$-Noetherian module. In this paper, we study some properties of right $S$-Noetherian rings and $S$-Noetherian modules. Among other things, we study Ore extensions, skew-Laurent polynomial ring extensions, and power series ring extensions of $S$-Noetherian rings.

Let $R$ be a commutative ring with identity. We use $\Gamma ( R )$ to denote the comaximal ideal graph. The vertices of $\Gamma ( R )$ are proper ideals of R that are not contained in the Jacobson radical of $R$, and two vertices $I$ and $J$ are adjacent if and only if $I + J = R$. In this paper we show some properties of this graph together with the planarity and perfection of $\Gamma ( R )$.

Let $\zeta_{n}$ be a primitive $n$th root of unity. As is well known, $\mathbf{Z}[\zeta_{n}+\zeta_{n}^{-1}]$ is the ring of integers of $\mathbf{Q}(\zeta_{n}+\zeta_{n}^{-1})$. We give an alternative proof of this fact by using the resultants of modified cyclotomic polynomials.

Rings all of whose right ideals are automorphism-invariant are called right [Formula: see text]-rings. In the present paper, we study rings having the property that every right cyclic module is dual automorphism-invariant. Such rings are called right [Formula: see text]-rings. We obtain some of the relationships [Formula: see text]-rings and [Formula: see text]-rings. We also prove that; (i) A semiperfect ring [Formula: see text] is a right [Formula: see text]-ring if and only if any right ideal in [Formula: see text] is a left [Formula: see text]-module, where [Formula: see text] is a subring of [Formula: see text] generated by its units, (ii) [Formula: see text] is semisimple artinian if and only if [Formula: see text] is semiperfect and the matrix ring [Formula: see text] is a right [Formula: see text]-ring for all [Formula: see text], (iii) Quasi-Frobenius right [Formula: see text]-rings are Frobenius.

We apply recent results on Galois-ring extensions and trace surjective algebras to analyze dehomogenized modular invariant rings of finite p-groups, as well as related localizations. We describe criteria for the dehomogenized invariant ring to be polynomial or at least regular and we show that for regular affine algebras with possibly non-linear action by a p-group, the singular locus of the invariant ring is contained in the variety of the transfer ideal. If V is the regular module of an arbitrary finite p-group, or V is any faithful representation of a cyclic p-group, we show that there is a suitable invariant linear form, inverting which renders the ring of invariants into a “localized polynomial ring” with dehomogenization being a polynomial ring. This is in surprising contrast to the fact that for a faithful representation of a cyclic group of order larger than p, the ring of invariants itself cannot be a polynomial ring by a result of Serre. Our results here generalize observations made by Richman [R] and by Campbell and Chuai [CCH].

We compute K-theory for ring C*-algebras in the case of higher roots of unity and thereby completely determine the K-theory for ring C*-algebras attached to rings of integers in arbitrary number fields., 26 pages, final version, accepted for publication in the Journal of Topology and Analysis

A ring R is called left p.q.-Baer if the left annihilator of a principal left ideal is generated, as a left ideal, by an idempotent. It is first proved that for a ring R and a group G, if the group ring RG is left p.q.-Baer then so is R; if in condition G is finite then |G|−1 ∈ R. Counterexamples are given to answer the question whether the group ring RG is left p.q.-Baer if R is left p.q.-Baer and G is a finite group with |G|−1 ∈ R. Further, RD∞ is left p.q.-Baer if and only if R is left p.q.-Baer.

A commutative local ring is generally defined to be a complete intersection if its completion is isomorphic to the quotient of a regular local ring by an ideal generated by a regular sequence. It has not previously been determined whether or not such a ring is necessarily itself the quotient of a regular ring by an ideal generated by a regular sequence. In this article, it is shown that if a complete intersection is a one dimensional integral domain, then it is such a quotient. However, an example is produced of a three dimensional complete intersection domain which is not a homomorphic image of a regular local ring, and so the property does not hold in general.

Let R be a prime ring, f(X 1, …, X n ) a multilinear polynomial which is not central-valued on R, and G a nonzero generalized skew derivation of R. Suppose that G(f(x 1, …, x n )) is zero or invertible for all x 1, …, x n ∈ R. Then it is proved that R is either a division ring or the ring of all 2 × 2 matrices over a division ring. This result simultaneously generalizes a number of results in the literature.

Let R be a commutative ring with identity. R is a finite factorization ring (FFR) if every nonzero nonunit of R has only a finite number of factorizations up to order and associates. In this article, we give a characterization of R for R[X] and R[[X]] to be an FFR.

Let α be an automorphism of a ring R. We study the skew Armendariz of Laurent series type rings (α-LA rings), as a generalization of the standard Armendariz condition from polynomials to skew Laurent series. We study on the relationship between the Baerness and p.p. property of a ring R and these of the skew Laurent series ring R[[x, x −1; α]], in case R is an α-LA ring. Moreover, we prove that for an α-weakly rigid ring R, R[[x, x −1; α]] is a left p.q.-Baer ring if and only if R is left p.q.-Baer and every countable subset of S l(R) has a generalized countable join in R. Various types of examples of α-LA rings are provided.

In this paper, we give a new formula of J:J¯ for any parameter ideal J in a Gorenstein local ring R of positive characteristic in terms of test ideals: J:J¯=J+τ(Jd−1), where τ(Jd−1) denotes the Jd−1-test ideal of R. As an application, we give a variant of Wangʼs theorem. Namely, we prove that if J is a parameter ideal in a Cohen–Macaulay local ring (R,m) of dimension d⩾2 with J⊆ms, then J:m(d−1)(s−1) (resp. J:m(d−1)(s−1)+1) is integral over J (resp. if R is not regular). Moreover, we prove that, after reduction to characteristic p≫0, a similar assertion holds true for Cohen–Macaulay Q-Gorenstein normal local domain essentially of finite type over a field of characteristic zero under some extra assumption.

Let R be a commutative ring with nonzero identity. In this paper, we study the von Neumann regular elements of R. We also study the idempotent elements, …- regular elements, the von Neumann local elements, and the clean elements of R. Finally, we investigate the subgraphs of the zero-divisor graph i(R) of R induced by the above elements. 2000 Mathematics Subject Classiflcation: primary 13F99; secondary 16E50, 13A15

It is proved that if R is a commutative ring in which zero is an adequate element, then R is an exchange ring and every zero adequate ring is an exchange ring. There is a new description of adequate rings; this is an answer to questions formulated by Larsen, Lewis, and Shores.

The notion of quasipolar elements of rings was introduced by Koliha and Patricio in 2002. In this paper, we introduce the notion of quasipolar rings and relate it to other familiar notions in ring theory. It is proved that both strongly π-regular rings and uniquely clean rings are quasipolar, and quasipolar rings are strongly clean, but no two of these classes of rings are equivalent. For commutative rings, quasipolar rings coincide with semiregular rings. It is also proved that every n × n upper triangular matrix ring over any commutative uniquely clean ring or commutative local ring is quasipolar.

A ring R is said to satisfy the Goodearl–Menal condition if for any x, y ∈ R, there exists a unit u of R such that both x − u and y − u −1 are units of R. It is proved that if R is a semilocal ring or an exchange ring with primitive factors Artinian, then R satisfies the Goodearl–Menal condition if and only if no homomorphic image of R is isomorphic to either ℤ2 or ℤ3 or 𝕄2(ℤ2). These results correct two existing results. Some consequences are discussed.

An element a of a ring R is called J-quasipolar if there exists p 2 = p ∈ R satisfying p ∈ comm2(a) and a + p ∈ J(R); R is called J-quasipolar in case each of its elements is J-quasipolar. The class of this sort of rings lies properly between the class of uniquely clean rings and the class of quasipolar rings. In particular, every J-quasipolar element in a ring is quasipolar. It is shown, in this paper, that a ring R is J-quasipolar iff R/J(R) is boolean and R is quasipolar. For a local ring R, we prove that every n × n upper triangular matrix ring over R is J-quasipolar iff R is uniquely bleached and R/J(R) ≅ ℤ2. Moreover, it is proved that any matrix ring of size greater than 1 is never J-quasipolar. Consequently, we determine when a 2 × 2 matrix over a commutative local ring is J-quasipolar. A criterion in terms of solvability of the characteristic equation is obtained for such a matrix to be J-quasipolar.

Let R be a commutative ring with Noetherian spectrum in which zero is a primary ideal. We determine the minimal zero-dimensional extensions of R when every regular prime ideal of R is contained in only finitely many prime ideals. This extends previous results of the first author for dim (R) ≤1. We also present a characterization of the partially ordered set of prime ideals in a ring with Noetherian spectrum.

If a (commutative unital) ring \(A\) is reduced and coincides with its total quotient ring, then \(A\) satisfies Property A (that is, \(A\) is a McCoy ring) if and only if the inclusion of \(A\) in its complete ring of quotients \(C(A)\) is a survival extension. The “if” assertion fails if one deletes the hypothesis that \(A\) is reduced. This is shown by using the idealization construction to construct a suitable ring \(A\) and then identifying its complete ring of quotients (which turns out to be a related idealization). Related characterizations of von Neumann regular rings are also given with the aid of the going-down property GD of ring extensions. For instance, a ring \(A\) is von Neumann regular if and only if \(A\) is a reduced McCoy ring that coincides with its total quotient ring such that \(A \subseteq C(A)\) satisfies GD.

The weakly regularity of all right R-modules with R an arbitrary ring does not imply the same property of all left R-modules. We describe the rings over which every right and left module is weakly regular and also obtain some description of semiartinian CSL-rings.

Let (R, m, k) be a normal Noetherian analytically irreducible universally catenary local domain of dimension d ≥ 1 with infinite residue field k and field of fractions K, and let I be an m-primary ideal. For P ∈ Min(m R[It]) there exists at least one such that P = Q ∩ R[It]. Let v be the Rees valuation of . We obtain necessary and sufficient conditions for the quotient ring R[It]/P to be a polynomial ring in d variables over k in terms of the v-values of a suitably chosen minimal generating set for I. Let J: = (a 1,…, a d )R be a minimal reduction of I. If I is a normal ideal and is a polynomial ring in d variables over k, we show that the residue field k(v) of V, is generated over k by the images of . If (R, m, k) is a two-dimensional regular local ring with algebraically closed residue field k and I is a product of distinct simple integrally closed m-primary ideals, we show for each positive integer n and each Q ∈ Min(m R[I n t]) that the ring is a two-dimensional normal Cohen–Macaulay graded domain wit...

An element a ∈ R is unit-regular provided that there exists an invertible u ∈ R such that a = aua. A ring R is called an almost unit-regular ring provided that for any a ∈ R, either a or 1 − a is unit-regular. We characterize, in this article, the almost unit-regularity of Morita contexts with zero pairings. We also show that a ring R is unit-regular if and only if M 2(R) is almost unit-regular. Various examples of such rings are constructed by means of formal triangular matrix rings.

Let m, n, r be nonzero fixed positive integers, R a 2-torsion free prime ring, Q its right Martindale quotient ring, and L a non-central Lie ideal of R. Let D : be a skew derivation of R and . We prove that if for all , then D is a usual derivation of R or R satisfies , the standard identity of degree 4.

In this paper, we extend the notion of a tip-extended pair of fuzzy subgroups to L-ideals of a ring. We prove that the sum of two tip-extended L-ideals of an arbitrary pair of L-ideals of a ring is the least L-ideal containing the union of the given L-ideals. Using this construction of join of L-ideals, we prove that the lattice of all L-ideals of a given ring is modular.

In this paper we obtain a characterization of prime and irreducible elements of the ring of integers modulo n. Also we give an explicit formula for computing the number of primes and irreducibles of this ring.

An element of the form [a, b, c] = abc − cba in a ring is called a generalized commutator. In this article, we show that, in a matrix ring 𝕄 n (S) (n ≥ 2) over any ring S (with identity), every matrix is a sum of a commutator and a generalized commutator. If S is an elementary divisor ring (in the sense that every square matrix over S is equivalent to a diagonal matrix), then every matrix in 𝕄 n (S) (n ≥ 2) is, in fact, a generalized commutator. Using suitable generic techniques, we show, however, that this conclusion is not true in general for various rings S (e.g. polynomial rings and power series rings). Indeed, for any n ≥ 2, there exists an n × n traceless matrix over some commutative ring S that is not a generalized commutator (respectively, is a generalized commutator but not a commutator) in 𝕄 n (S). This gives, in particular, a strong negative solution to the problem whether n × n traceless matrices are necessarily commutators over a commutative base ring, for any n ≥ 2.

Given a ring R, we define its right i-profile (resp. right p-profile) to be the collection of injectivity domains (resp. projectivity domains) of its right R-modules. We study the lattice theoretic properties of these profiles and consider ways in which properties of the profiles may determine the structure of rings and viceversa. We show that the i-profile is isomorphic to an interval of the lattice of linear filters of right ideals of R, and is therefore modular and coatomic. In particular, we give a practical characterization of the i-profile of a right artinian ring. We show through an example that the p-profile is not necessarily a set, and also characterize the right p-profile of a right perfect ring. The study of rings in terms of their (i- or p-)profile was inspired by the study of rings with no (i- or p-) middle class, initiated in recent papers by Er, L\'opez-Permouth and S\"okmez, and by Holston, L\'opez-Permouth and Orhan-Ertas. In this paper, we obtain further results about these rings and we also use our results to provide a characterization of a special class of QF-rings in which the injectivity and projectivity domains of any module coincide., Comment: 19 pages, examples and propositions added. Title changed

Let 𝒞 be a pocategory, (see Khripchenko [3]), and FI(𝒞) its finitary incidence ring [3]. We prove that the ring ODer FI of outer derivations of FI(𝒞) is isomorphic to the ring ODer 𝒞 of outer derivations of 𝒞. Furthermore, if 𝒞 = 𝒞(P, R), where P is a quasiordered set and R is a ring, then it is proved that . As a consequence, the description of the algebra K-ODer FI of outer K-derivations of FI(P, A), where A is a K-algebra, is obtained. The ring of outer derivations and the algebra of outer K-derivations of the weak incidence ring I*(𝒞) [3] are also described.

We consider a ring whose left modules of finite length are semisimple and prove some results on such a ring. We also consider when such a ring is a left V-ring.

Let R be a ring with identity, and let M be a unitary (left) R-module. Then M is said to be torsion provided that for every m ∈ M, there is a nonzero r ∈ R such that rm = 0. In this article, we study the question of the existence (and nonexistence) of faithful torsion modules over both commutative and noncommutative rings.

In this paper, we concern the w-analogue of Matijevic’s result. We show that if R is a w-Noetherian ring and T a w-overring of R such that T ⊆ Rwg. Then T has ACC on regular w-ideals.

A ring is called semiregular if is regular and idem-potents lift modulo , where denotes the Jacobson radical of . We give some characterizations of rings such that idempotents lift modulo , and satisfies one of the following conditions: (one-sided) unit-regular, strongly regular, (unit, strongly, weakly) -regular.

Filial rings are rings in which the relation of being an ideal is transitive. We continue the study of commutative filial rings started in [2, 3]. In particular, we give a complete description of the commutative, filial, noetherian rings with identity.

In this article we prove some results about lifting of generators of ideal to the Laurent polynomial ring analogous to the polynomial ring.

Let A be the adjacency matrix of the zero-divisor graph Γ ( R ) of a finite commutative ring R containing nonzero zero-divisors. In this paper, it is shown that Γ ( R ) is the zero-divisor graph of a Boolean ring if and only if det ( A ) = - 1 . Also, A is similar to plus or minus its inverse whenever R is a Boolean ring. As a consequence, it is proved that Γ ( R ) is the zero-divisor graph of a Boolean ring if and only if the set of eigenvalues (including multiplicities) of Γ ( R ) can be partitioned into 2-element subsets of the form { λ , ± 1 / λ } . Furthermore, any finite Boolean ring R is characterized by the degree and coefficients of the characteristic polynomial of A.

Mason introduced the reflexive property for ideals, and then this concept was generalized by Kim and Baik, defining idempotent reflexive right ideals and rings. In this article, we characterize aspects of the reflexive and one-sided idempotent reflexive properties, showing that the concept of idempotent reflexive ring is not left-right symmetric. It is proved that a (right idempotent) reflexive ring which is not semiprime (resp., reflexive), can always be constructed from any semiprime (resp., reflexive) ring. It is also proved that the reflexive condition is Morita invariant and that the right quotient ring of a reflexive ring is reflexive. It is shown that both the polynomial ring and the power series ring over a reflexive ring are idempotent reflexive. We obtain additionally that the semiprimeness, reflexive property and one-sided idempotent reflexive property of a ring coincide for right principally quasi-Baer rings.

An element of a ring is called strongly J-clean provided that it can be written as the sum of an idempotent and an element in its Jacobson radical that commute. We investigate, in this article, a single strongly J-clean 2 × 2 matrix over a noncommutative local ring. The criteria on strong J-cleanness of 2 × 2 matrices in terms of a quadratic equation are given. These extend the corresponding results in [8, Theorems 2.7 and 3.2], [9, Theorem 2.6], and [11, Theorem 7].

We consider the group G of R-automorphisms of the polynomial ring R[x] in the case where the ring R has nonzero nilpotent elements. Little is known about G in this case, and because of the importance of G in understanding questions involving the polynomial ring R[x], we initiate here several lines of investigation. We do this by examining in detail examples involving the ring of integers modulo n. If R is a local ring with maximal ideal m such that R/m = ℤ2 and m 2 = (0), we describe more explicitly the structure of G and determine all rings of invariants of R[x] with respect to subgroups of G.

We generalize the insertion-of-factors-property by setting nil- potent products of elements. In the process we introduce the concept of a nil-IFP ring that is also a generalization of an NI ring. It is shown that if Kothe's conjecture holds, then every nil-IFP ring is NI. The class of minimal noncommutative nil-IFP rings is completely determined, up to isomorphism, where the minimal means having smallest cardinality.

This paper aims to show that the “going-down ring” and the “divided ring” properties ascend along flat morphisms whose co-diagonal morphisms are flat, the so-called absolutely flat morphisms introduced by Olivier. But unibranchedness hypotheses are necessary as any henselization morphism shows. As a by-product, we get that the “unibranched divided ring” property is preserved by the formation of factor domains and by localization with respect to prime ideals. Moreover, we exhibit some ascent results for the “going-down ring” and “divided ring” properties along integral extensions. Open image in new window

It is shown that if a lattice-ordered n × n (n ≥ 2) matrix ring over a totally ordered integral domain or division ring containing a positive n-cycle, then it is isomorphic to the lattice-ordered n × n matrix ring with entrywise lattice order.

For a ring R with a derivation δ, we introduce and investigate a generalization of reduced rings which we call an Armendariz ring of pseudo-differential operator type (or simply 𝒟𝒪-Armendariz ring). Various classes of non-reduced 𝒟𝒪-Armendariz rings is provided and a number of properties of this generalization are established. Radicals of the pseudo-differential operator ring R((x −1, δ)), in terms of those of a 𝒟𝒪-Armendariz ring R, is established.

This paper establishes some interrelations between a finite sequence of sets of mappings of an abstract set S into complete residue systems of pairwise relatively prime elements of any Dedekind ring and the corresponding sequence of sets of mappings of the set S into the complete residue system corresponding to the product of these elements. A relationship is revealed between the established interrelation and Lang's theorem on isomorphic factor rings. A string model is presented that is an interpretation of structures investigated in the cases of the ring of integers and a one-element set S. It is shown that the results obtained can be used for computing the number of combinatorial objects defined in terms of finite residue rings.

In this paper we introduce two operators denoted by ( ) ( ) n and ( ) u of a differential ring constructed from a subset of a differential ring. We shall also discuss the relationship between these operators and the differential ideals in differential rings, and Keigher differential ring. n A a a A   for

Let R be a commutative ring with identity. A nonunit element a ∈ R is an atom if a = bc (b, c ∈ R) implies (a) = (b) or (a) = (c), and R is atomic if each nonunit of R is a finite product of atoms. A ring R is a (weak) CK ring if R is atomic and (each maximal ideal of) R contains only finitely many nonassociate atoms. We show that the following are equivalent: (1) R is a weak CK ring, (2) each (prime) ideal of R is a finite union of principal ideals, (3) R is atomic and each maximal ideal is a finite union of principal ideals, and (4) R is a finite direct product of finite local rings, special principal ideal rings (SPIRs), and (one-dimensional) Noetherian domains in which every maximal ideal is a finite union of principal ideals (or equivalently, weak CK domains). We show that a domain R is a weak CK domain if and only if R is a Noetherian domain with R M a CK domain for each maximal ideal M of R and Pic(R) = 0.