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Given a commutative ring with identity [Formula: see text] and an [Formula: see text]-module [Formula: see text], a subset [Formula: see text] of [Formula: see text] is a cyclic covering of [Formula: see text], if this module is the union of the cyclic submodules [Formula: see text], where [Formula: see text]. Such covering is said to be irredundant, if no proper subset of [Formula: see text] is a cyclic covering of [Formula: see text]. In this work, an irredundant cyclic covering of [Formula: see text] is constructed for every Artinian commutative ring [Formula: see text]. As a consequence, a cyclic covering of minimal cardinality of [Formula: see text] is obtained for every finite commutative ring [Formula: see text], extending previous results in the literature.

We study the structure of rings whose principal right ideals contain a sort of two-sided ideals, introducing right �-duo as a generaliza- tion of (weakly) right duo rings. Abelian �-regular rings are �-duo, which is compared with the fact that Abelian regular rings are duo. For a right �-duo ring R, it is shown that every prime ideal of R is maximal if and only if R is a (strongly) �-regular ring with J(R) = N∗(R). This result may be helpful to develop several well-known results related to pm rings (i.e., rings whose prime ideals are maximal). We also extend the right �-duo property to several kinds of ring which have roles in ring theory. Throughout this note every ring is associative with identity unless otherwise specified. Given a ring R (possibly without identity), J(R), N∗(R), N ∗ (R), and N(R) denote the Jacobson radical, the prime radical, the upper nilradical (i.e., sum of all nil ideals), and the set of all nilpotent elements in R, respectively. It is well-known that N ∗ (R) ⊆ J(R) and N∗(R) ⊆ N ∗ (R) ⊆ N(R). We use R(x) (R((x))) to denote the polynomial (power series) ring with an indeterminate x over R. Denote the n by n full (resp., upper triangular) matrix ring over R by Matn(R) (resp., Un(R)). Use eij for the matrix unit with (i,j)-entry 1 and else- where 0. Denote {(aij) ∈ Un(R) | the diagonal entries of (aij) are all equal} by Dn(R). rR(−) (resp., lR(−)) is used to denote a right (resp., left) annihi- lator in R. Q denotes the direct product of rings. Z (Zn) denotes the ring of integers (modulo n). 1. Right �-duo rings In this section we introduce the concept of a right π-duo ring as a general- ization of weakly right duo ring, and study the structure of right π-duo rings. Let R be a ring and M be a right R-module. Buhphang and Rege (4) called M semicommutative if mRa = 0 whenever ma = 0 for m ∈ M and a ∈ R. We first consider the condition (∗): If ma = 0 for m ∈ M and a ∈ R, then mRa n = 0 for some n ≥ 1

As a consequence of Cohen's structure Theorem for complete local rings that every _nite commutative ring R of characteristic pn contains a unique special primary subring R0 satisfying R/J(R) = R0/pR0: Cohen called R0 the coe_cient subring of R. In this paper we will study the case when the ring is a transcendental extension local artinian duo ring R; we proved that even in this case R will has a commutative coe_cient subring.

Let R be a commutative ring, Q0(R) be the ring of finite fractions over R, and w be the so-called w-operation on R. In this article, we introduce a new type of Prufer v-multiplication ring, called a quasi-Q0-PvMR and defined as a ring R for which every w-linked Q0-overring of R is integrally closed in Q0(R). Our primary motivation for investigating quasi-Q0-PvMRs is to provide w-theoretic analogues to some work of Lucas [16] concerning Q0-Prufer rings. (A ring R is called a Q0-Prufer ring if every Q0-overring of R is integrally closed in Q0(R).)

A ring R is called a ring with a large center or an IIC-ring if any nonzero ideal of R has a nonzero intersection with the center of R. We consider conditions which guarantee that a semigroup ring over an IIC-ring is an IIC-ring.

The left quotient ring (i.e. the left classical ring of fractions) $Q_{cl}(R)$ of a ring $R$ does not always exist and still, in general, there is no good understanding of the reason why this happens. In this paper, it is proved existence of the largest left quotient ring $Q_l(R)$, i.e. $Q_l(R) = S_0(R)^{-1}R$ where $S_0(R)$ is the largest left regular denominator set of $R$. It is proved that $Q_l(Q_l(R))=Q_l(R)$; the ring $Q_l(R)$ is semi-simple iff $Q_{cl}(R)$ exists and is semi-simple; moreover, if the ring $Q_l(R)$ is left artinian then $Q_{cl}(R)$ exists and $Q_l(R) = Q_{cl}(R)$. The group of units $Q_l(R)^*$ of $Q_l(R)$ is equal to the set $\{s^{-1} t\, | \, s,t\in S_0(R)\}$ and $S_0(R) = R\cap Q_l(R)^*$. If there exists a finitely generated flat left $R$-module which is not projective then $Q_l(R)$ is not a semi-simple ring. We extend slightly Ore's method of localization to localizable left Ore sets, give a criterion of when a left Ore set is localizable, and prove that all left and right Ore sets of an arbitrary ring are localizable (not just denominator sets as in Ore's method of localization). Applications are given for certain classes of rings (semi-prime Goldie rings, Noetherian commutative rings, the algebra of polynomial integro-differential operators)., Comment: 32 pages

In this paper we examine the commutativity of ideal extensions. We introduce methods of constructing such extensions, in particular we construct a noncommutative ring T which contains a central and idempotent ideal I such that T/I is a field. This answers a question from [2]. Moreover we classify fields of characteristic 0 which can be obtained as T/I for some T., Comment: 7 pages

We identify the spaces of homogeneous polynomials in two variables 𝕂[Yk, XYk−1, ⋅, Xk] among representations of the Lie ring 𝔰𝔩2(𝕂). This amounts to constructing a compatible 𝕂-linear structure on some abstract 𝔰𝔩2(𝕂)-modules, where 𝔰𝔩2(𝕂) is viewed as a Lie ring.

Let $K$ be an associative and commutative ring with $1$, $k$ a subring of $K$ such that $1\in k$, $n\geq 2$ an integer. The paper describes subrings of the general linear Lie ring $gl_{n} ( K )$ that contain the Lie ring of all traceless matrices over $k$.

Let R be a prime ring, extended centroid C, Utumi quotient ring U, and m, n ≥ 1 are fixed positive integers, F a generalized derivation associated with a nonzero derivation d of R. We study the case when one of the following holds: (i) F ( x ) ∘ m d ( y ) = ( x ∘ y ) n and (ii) ( F ( x ) ∘ d ( y ) ) m = ( x ∘ y ) n , for all x, y in some appropriate subset of R. We also examine the case where R is a semiprime ring.

In this study, we consider various graphs with the vertices being sets of ideals of a commutative ring with identity, R. How we define the edge set will vary according to distinct ideal theoretic properties. We apply a graphical approach to the study of ideal relationships in commutative rings and from this we obtain some properties of their graphs.

Let R be a finite principal ideal ring and m , n , d positive integers. In this paper, we study the matrix graph over R which is the graph whose vertices are m × n matrices over R and two matrices A and B are adjacent if and only if 0 rank ( A − B ) d . We show that this graph is a connected vertex transitive graph. The distance, diameter, independence number, clique number and chromatic number of this graph are also determined. This graph can be applied to study MRD codes over R. We obtain that a maximal independent set of the matrix graph is a maximum rank distance (MRD) code and vice versa. Moreover, we show the existence of linear MRD codes over R.

In this paper, a necessary and sufficient condition for the homogeneous distance on an arbitrary finite commutative principal ideal ring to be a metric is obtained. We completely characterize the lower bound of homogeneous distances of matrix product codes over any finite principal ideal ring where the homogeneous distance is a metric. Furthermore, the minimum homogeneous distances of the duals of such codes are also explicitly investigated.

We prove that for every trace zero matrix $A$ over a principal ideal ring $R$, there exist trace zero matrices $X,Y$ over $R$ such that $XY-YX=A$. Moreover, we show that $X$ can be taken to be regular mod every maximal ideal of $R$. This strengthens our earlier result that $A$ is a commutator of two matrices (not necessarily of trace zero), and in addition, the present proof is significantly simpler than the earlier one. Shalev has conjectured an analogous statement for group commutators in $\mathrm{SL}_{n}$ over $p$-adic integers. We prove Shalev's conjecture for $n=2$., Comment: 15 pages; Lemma 2.3 and Theorem 5.3 have new corrected proofs; local-global principle emphasised

The rings considered in this article are commutative with identity and which are not integral domains. The aim of this article is to classify zero-dimensional semiquasilocal rings with at least two maximal ideals such that their annihilating ideal graphs are planar.

Let R be a ring, (S, ≤) a strictly ordered monoid and ω: S → End(R) a monoid homomorphism. The skew generalized power series ring R[[S, ω]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal'cev–Neumann Laurent series rings. In this article, we study relations between the (quasi-) Baer, principally quasi-Baer and principally projective properties of a ring R, and its skew generalized power series extension R[[S, ω]]. As particular cases of our general results, we obtain new theorems on (skew) group rings, Mal'cev–Neumann Laurent series rings, and the ring of generalized power series.

The content of a polynomial f over a commutative ring R is the ideal c(f) of R generated by the coefficients of f. A commutative ring R is said to be Gaussian if c(fg) = c(f)c(g) for every polynomials f and g in R[X]. A number of authors have formulated necessary and sufficient conditions for R(X) (respectively, R⟨X⟩) to be semihereditary, have weak global dimension at most one, be arithmetical, or be Prufer. An open question raised by Glaz is to formulate necessary and sufficient conditions that R(X) (respectively, R⟨X⟩) have the Gaussian property. We give a necessary and sufficient condition for the rings R(X) and R⟨X⟩ in terms of the ring R in case the square of the nilradical of R is zero.

According to Jacobson (31), a right ideal is bounded if it con- tains a non-zero ideal, and Faith (15) called a ring strongly right bounded if every non-zero right ideal is bounded. From (30), a ring is strongly right AB if every non-zero right annihilator is bounded. In this paper, we introduce and investigate a particular class of McCoy rings which sat- isfy Property (A) and the conditions asked by Nielsen (42). It is shown that for a u.p.-monoid M and � : M ! End(R) a compatible monoid homomorphism, if R is reversible, then the skew monoid ring RM is strongly right AB. If R is a strongly right AB ring, M is a u.p.-monoid and � : M ! End(R) is a weakly rigid monoid homomorphism, then the skew monoid ring RM has right Property (A).

This note is concerned with examining nilradicals and Jacob-son radicals of polynomial rings when related factor rings are Armendariz.Especially we elaborate upon a well-known structural property of Armen-dariz rings, bringing into focus the Armendariz property of factor rings byJacobson radicals. We show that J(R[x]) = J(R)[x] if and only if J(R) isnil when a given ring R is Armendariz, where J(A) means the Jacobsonradical of a ring A. A ring will be called feckly Armendariz if the factorring by the Jacobson radical is an Armendariz ring. It is shown that thepolynomial ring over an Armendariz ring is feckly Armendariz, in spiteof Armendariz rings being not feckly Armendariz in general. It is alsoshown that the feckly Armendariz property does not go up to polynomialrings. 1. On radicals whenfactor rings are ArmendarizThroughout this note every ring is associative with identity unless other-wise stated. For a ring R, J(R), N ∗ (R), N ∗ (R), N 0 (R) and N(R) denotethe Jacobson radical, the prime radical, the upper nilradical (i.e., sum of allnil ideals), the Wedderburn radical (i.e., the sum of all nilpotent ideals), andthe set of all nilpotent elements in R, respectively. Following [1, p. 130], asubset of R is said to be locally nilpotent if its finitely generated subringsare nilpotent. Also due to [1, p. 130], the Levitzki radical of R, written bysσ(R), means the sum of all locally nilpotent ideals of R. It is well-knownthat N

We prove, in this note, that a Zabavsky ring R is an elementary divisor ring if and only if R is a ring. Many known results are thereby generalized to much wider class of rings, e.g. [4, Theorem 14], [7, Theorem 4], [9, Theorem 1.2.14], [11, Theorem 4] and [12, Theorem 7].

Let A and B be commutative rings with unity, f: A → B a ring homomorphism, and J an ideal of B. Then the subring A ⋈fJ: = {(a, f(a) + j)|a ∈ A and j ∈ J} of A × B is called the amalgamation of A with B along J with respect to f. In this article, among other things, we investigate the Cohen–Macaulay and (quasi-)Gorenstein properties on the ring A ⋈fJ.

We express the double affine Hecke algebra ℍ associated to the general linear group GL2(k) (here, k is a field with char(k) ≠ 2) as an amalgamated free product of quadratic extensions over the three-dimensional quantum torus 𝒪q((k×)3). With an eye towards proving ring-theoretic results pertaining to ℍ, a general treatment of amalgamated products of Ore and quadratic extensions is given. We prove an analogue of the Hilbert Basis Theorem for an amalgamated product Q of quadratic extensions and determine conditions for when the one-sided ideals of Q are principal or doubly-generated. Furthermore, we determine sufficient conditions which imply Q is a principal ideal ring. Finally, we construct an explicit isomorphism from ℍ to the amalgamated free product ring of quadratic extensions over 𝒪q((k×)3), a ring known to be noetherian. Therefore, it follows that ℍ is noetherian.

It is proved that a commutative ring A is arithmetical if and only if every finitely generated ideal M of the ring A is a quasi-projective A-module and every endomorphism of this module can be extended to an endomorphism of the module AA. These results are proved with the use of some general results on invariant arithmetical rings.

In this paper, we give examples of infinite series of finite rings B () , where m ≥ 2, 0 ≤ v ≤ p−1, and p is a prime number, that are not representable by matrix rings over commutative rings, and we describe the basis of polynomial identities of these rings. We prove here that every variety var B () , where m = 2 or m − 1 = (p − 1)k, k ≥ 1, and p ≥ 3 or p = 2 and m ≥ 3, 0 ≤ v < p, and p is a prime number, is a minimal variety containing a finite ring that is nonrepresentable by a matrix ring over a commutative ring. Therefore, we describe almost finitely representable varieties of rings whose generating ring contains an idempotent element of additive order p.

In this paper, we introduce the concept of completely arithmetical rings and investigate their properties. In particular, we prove that if R is a completely arithmetical ring with J(R)=0, then K0(R) ≅ ℤn for some positive integer n. We also show that such a ring is precisely a ring in which every proper ideal can be written uniquely as a product of finitely many distinct completely strongly irreducible ideals.

As a proper generalization of injective modules in term of supplements, we say that a module M has the property (SE) (respectively, the property (SSE)) if, whenever M ( N, M has a supplement that is a direct summand of N (respectively, a strong supplement in N). We show that a ring R is a left and right artinian serial ring with Rad(R)2 = 0 if and only if every left R-module has the property (SSE). We prove that a commutative ring R is an artinian serial ring if and only if every left R-module has the property (SE).

We make a study of two generalizations of regular rings, concentrating our attention on the structure of idempotents. A ring R is said to be right attaching-idempotent if for there exists such that ab is an idempotent. Next R is said to be generalized regular if for there exist nonzero such that ab is a nonzero idempotent. It is first checked that generalized regular is left-right symmetric but right attaching-idempotent is not. The generalized regularity is shown to be a Morita invariant property. More structural properties of these two concepts are also investigated.

Let R be a commutative ring with identity. Let A(R) denote the collection of all annihilating ideals of R (that is, A(R) is the collection of all ideals I of R which admits a nonzero annihilator in R). Let AG(R) denote the annihilating ideal graph of R. In this article, necessary and sufficient conditions are determined in order that AG(R) is complemented under the assumption that R is a zero-dimensional quasisemilocal ring which admits at least two nonzero annihilating ideals and as a corollary we determine finite rings R such that AG(R) is complemented under the assumption that A(R) contains at least two nonzero ideals.

This paper mainly studies some properties of skew polynomial ring related to Morita invariance, Armendariz and (quasi)-Baer. First, we show that skew polynomial ring has no Morita invariance by the counterexample. Then we prove a necessary condition that skew polynomial ring constitutes Armendariz ring. We lastly investigate that condition of skew polynomial ring is a (quasi)-Baer ring, and verify that the conditions is necessary, but not sufficient by example and counterexample.

We combine Euclidean and adequate rings, and introduce a new type of ring. A ring $R$ is called an E-adequate ring provided that for any $a,b\in R$ such that $aR+bR=R$ and $c\neq 0$ there exists $y\in R$ such that $(a+by,c)$ is an E-adequate pair. We shall prove that an E-adequate ring is an elementary divisor ring if and only if it is a Hermite ring. Elementary matrix reduction over such rings is also studied. We thereby generalize Domsha, Vasiunyk, and Zabavsky's theorems to a much wider class of rings.

We study the class of almost uniserial rings as a straightforward common generalization of left uniserial rings and left principal ideal domains. A ring R is called almost left uniserial if any two non-isomorphic left ideals of R are linearly ordered by inclusion, i.e., for every pair I, J of left ideals of R either I ⊆ J , or J ⊆ I , or I ≅ J . Also, an R-module M is called almost uniserial if any two non-isomorphic submodules are linearly ordered by inclusion. We give some interesting and useful properties of almost uniserial rings and modules. It is shown that a left almost uniserial ring is either a local ring or its maximal left ideals are cyclic. A Noetherian left almost uniserial ring is a local ring or a principal left ideal ring. Also, a left Artinian principal left ideal ring R is almost left uniserial if and only if R is left uniserial or R = M 2 ( D ) , where D is a division ring. Finally we consider Artinian commutative rings which are almost uniserial and we obtain a structure theorem for these rings.

Let R be a commutative ring with total quotient ring K. Each monomorphic R-module endomorphism of a cyclic R-module is an isomorphism if and only if R has Krull dimension 0. Each monomorphic R-module endomorphism of R is an isomorphism if and only if R = K. We say that R has property () if for each nonzero element , each monomorphic R-module endomorphism of R/Ra is an isomorphism. If R has property (), then each nonzero principal prime ideal of R is a maximal ideal, but the converse is false, even for integral domains of Krull dimension 2. An integral domain R has property () if and only if R has no R-sequence of length 2; the "if" assertion fails in general for non-domain rings R. Each treed domain has property (), but the converse is false.

We call a commutative ring R an F IN -ring (resp., F SA-ring) if for any two finitely generated I, J ⊴R we have Ann(I)+Ann(J )=Ann(I∩J ) (resp., there is K ⊴ R such that Ann(I)+Ann(J )=Ann(K)). Moreover, we extend this concepts to αIN -rings and αSA-rings where α is a cardinal number. The class of F SA-rings includes the class of all SA-rings (hence all IN -rings) and all P P -rings (hence all Baer-rings). In this paper, after giving some properties of αSA-rings, we prove that a reduced ring R is αSA if and only if it is an αIN -ring. Consequently, C(X) is an F SA-ring if and only if C(X) is an F IN -ring and equivalently X is an F -space. Moreover, for a commutative ring R, we have shown that R is a Baer-ring if and only if R is a reduced IN -ring. A topological space X is said to be an αU E-space if the closure of any union with cardinal number less than α of clopen subsets is open. Topological properties of αU E-spaces are investigated. Finally, we show that a completely regular Hausdorff space ...

Order complex is an important object associated to a partially ordered set. Following a suggestion from V. A. Vassiliev (1994), we investigate an order complex associated to the partially ordered set of nontrivial ideals in a commutative ring with identity. We determine the homotopy type of the geometric realization for the order complex associated to a general commutative ring with identity. We show that this complex is contractible except for semilocal rings with trivial Jacobson radical when it is homotopy equivalent to a sphere.

A ring R is a left AIP-ring if the left annihilator of any ideal of R is pure as a left ideal. Equivalently, R is a left AIP-ring if R modulo the left annihilator of any ideal is flat. This class of rings includes both right PP-rings and right p.q.-Baer rings (and hence the biregular rings) and is closed under direct products and forming upper triangular matrix rings. It is shown that, unlike the Baer or right PP conditions, the AIP property is inherited by polynomial extensions and has the advantage that it is a Morita invariant property. We also give a complete characterization of a class of AIP-rings which have a sheaf representation. Connections to related classes of rings are investigated and several examples and counterexamples are included to illustrate and delimit the theory.

In this paper we combine the concepts of Burnside ring and character ring in what we call the global representation ring of a finite group. We compute all ring homomorphisms from this ring to the complex numbers, we compute its spectrum, its connected components, and determine the primitive idempotents.

An associative ring R is called a unique addition ring (UA-ring) if its multiplicative semigroup (R, · ) can be equipped with a unique binary operation+ transforming the triple (R, ·, +) to a ring. An R-module A is said to be an End-UA-module if the endomorphism ring EndR(A) of A is a UA-ring. In the paper, the torsion-free End-UA-modules over commutative Dedekind domains are studied. In some classes of Abelian torsion-free groups, the Abelian groups having UA-endomorphism rings are found.

In this paper, we investigate the insertion-of-factors-proper- ty (simply, IFP) on skew polynomial rings, introducing the concept of strongly �-IFP for a ring endomorphism �. A ring R is said to have strongly �-IFP if the skew polynomial ring R(x;�) has IFP. We examine some characterizations and extensions of strongly �-IFP rings in relation with several ring theoretic properties which have important roles in ring theory. We also extend many of related basic results to the wider classes, and so several known results follow as consequences of our results. Throughout this paper, all rings are associative with identity. We denote by R(x) the polynomial ring with an indeterminate x over R. degf(x) denotes the degree of f(x) ∈ R(x). Let Z and Zn denote the ring of integers and the ring of integers modulo n, respectively. Due to Bell (3), a ring R is called to satisfy the insertion-of-factors-property (simply, an IFP ring) if ab = 0 implies aRb = 0 for a,b ∈ R. Narbonne (22) and Shin (26) used the terms semicommutative and SI for the IFP, respectively. Commutative rings have clearly IFP, and any reduced ring (i.e., a ring without nonzero nilpotent elements) has IFP by a simple computation. A ring is called Abelian if every idempotent is central. IFP rings are Abelian by a simple computation. Another generalization of a reduced ring is an Armendariz ring. Rege and Chhawchharia (25) called a ring R Armendariz if whenever the product of any two polynomials over a ring is zero, then so is the product of any pair of coefficients from the two polynomials. The class of IFP rings and the c of Armendariz rings do not imply each other by (25, Example 3.2) and (13, Example 14). Note that the polynomial rings over IFP rings need not have IFP by (13, Example 2), but for an Armendariz ring R, R has IFP if and only if

A ring R is called right Johns if R is right noetherian and every right ideal of R is a right annihilator. R is called strongly right Johns if the matrix ring M n (R) is right Johns for each integer n ≥ 1. The Faith–Menal conjecture is an open conjecture on QF rings. It says that every strongly right Johns ring is QF. It is proved that the conjecture is true if every closed left ideal of the ring R is finitely generated. This result improves the known result that the conjecture is true if R is a left CS ring.

In this article we investigate: conditions under which a primitive idempotent in a ring is central; the set of right ideals eR where e is a primitive idempotent in a ring R; the set of primitive idempotent ideals of a ring; and idempotent elements in the ring of matrices.

Hirano studied the quasi-Armendariz property of rings, and then this concept was generalized by some authors, defining quasi-Armendariz property for skew polynomial rings and monoid rings. In this article, we consider unified approach to the quasi-Armendariz property of skew power series rings and skew polynomial rings by considering the quasi-Armendariz condition in mixed extension ring [R; I][x; σ], introducing the concept of so-called (σ, I)-quasi Armendariz ring, where R is an associative ring equipped with an endomorphism σ and I is an σ-stable ideal of R. We study the ring-theoretical properties of (σ, I)-quasi Armendariz rings, and we obtain various necessary or sufficient conditions for a ring to be (σ, I)-quasi Armendariz. Constructing various examples, we classify how the (σ, I)-quasi Armendariz property behaves under various ring extensions. Furthermore, we show that a number of interesting properties of an (σ, I)-quasi Armendariz ring R such as reflexive and quasi-Baer property transfer to its...

Let R be a ring with a derivation δ and R((x-1; δ)) denote the pseudo-differential operator ring over R. We study the relations between the set of annihilators in R and the set of annihilators in R((x-1; δ)). Among applications, it is shown that for an Armendariz ring R of pseudo-differential operator type, the ring R((x-1; δ)) is Baer (resp., quasi-Baer, PP, right zip) if and only if R is a Baer (resp., quasi-Baer, PP, right zip) ring. For a δ-weakly rigid ring R, R((x-1; δ)) is a left p.q.-Baer ring if and only if R is left p.q.-Baer and every countable subset of left semicentral idempotents of R has a generalized countable join in R.

Let D be an integral domain, t be the so-called t-operation on D; and S be a (not necessarily saturated) multiplicative subset of D. In this paper, we study the Nagata ring of S-Noetherian domains and locally S-Noetherian domains. We also inves- tigate the t-Nagata ring of t-locally S-Noetherian domains. In fact, we show that if S is an anti-archimedean subset of D, then D is an S-Noetherian domain (respectively, locally S-Noetherian domain) if and only if the Nagata ring D(X)N is an S-Noetherian domain (respectively, locally S-Noetherian domain). We also prove that if S is an anti-archimedean subset of D, then D is a t-locally S-Noetherian domain if and only if the polynomial ring D(X) is a t-locally S-Noetherian domain, if and only if the t-Nagata ring D(X)Nv is a t-locally S-Noetherian domain.

We give a description of rings of formalmatrices belonging to one of the following classes of rings: semiartinian rings, max-rings, V -rings, SV -rings. We study semiartinian SSP-rings and SSP-rings of formal matrices.

A ring is a UMP-ring if every maximal ideal in the ring is the union of the minimal prime ideals it contains. Banerjee, Ghosh and Henriksen have characterized Tychonoff spaces X for which C(X) is a UMP-ring. One of the characterizations is that every singleton of β X is what is called a nearly round subset. In this article, we define nearly round quotient maps, and use them to characterize completely regular frames L for which $${\mathcal{R} L}$$ is a UMP-ring. All such frames are almost P-frames, and an Oz-frame is of this kind precisely when it is an almost P-frame. If L is perfectly normal (and hence if L is metrizable), then $${\mathcal{R} L}$$ is a UMP-ring if and only if L is Boolean. If A is a UMP-ring which is a $${\mathbb{Q}}$$ -algebra, then every ideal of A, when viewed as a ring in its own right, is a UMP-ring.

We consider the factor ring $$S = k[x; \sigma]/\langle x^{n} \rangle$$ , where k is a finite field. We show that this local ring can be represented by a matrix ring and any module over S is a direct sum of cyclic submodules. The ring S appears in Problem 2.45 of Dniester Notebook. For two such rings S 1 and S 2 we deal with left-right (S 1, S 2) bimodules. A certain factor ring of a skew polynomial ring over a Galois ring is also considered. This factor ring is isomorphic to a full matrix ring over some Galois ring.

Let R be a ring, and let S be a strictly ordered monoid. The generalized power series ring R[[S]] is a common generalization of polynomial rings, Laurent polynomial rings, power series rings, Laurent series rings, Mal'cev–Neumann series rings, monoid rings and group rings. In this paper, we examine which conditions on R and S are necessary and which are sufficient for the generalized power series ring R[[S]] to be semilocal right Bézout or semilocal right distributive. In the case where S is a strictly totally ordered monoid we characterize generalized power series rings R[[S]] that are semilocal right distributive or semilocal right Bézout (the latter under the additional assumption that S is not a group).

In this paper, we introduce and study the w-weak globaldimension w-w.gl.dim(R) of a commutative ring R. As an application, itis shown that an integral domain R is a Pru¨fer v-multiplication domainif and only if w-w.gl.dim(R) 6 1. We also show that there is a largeclass of domains in which Hilbert’s syzygy Theorem for the w-weak globaldimension does not hold. Namely, we prove that if Ris an integral domain(but not a field) for which the polynomial ring R[x] is w-coherent, thenw-w.gl.dim(R[x]) = w-w.gl.dim(R). 1. IntroductionRecall that a Pru¨fer domain is an integral domain in which every finitelygenerated ideal is invertible. It is well-known that the concept of Pru¨fer do-mains has played a central role in the development of the classical ideal theory.An important generalization of the Pru¨fer domain notion is that of a Pru¨ferv-multiplication domain (PVMD). This notion comes from multiplicative idealtheory, and various ideal-theoretic properties of it have been considered bymany authors, see for example [1, 2, 4, 7, 9, 10, 13, 14, 15]. From the homolog-ical algebra point of view, Pru¨fer domains are exactly the integral domains ofweak global dimension at most one. The original motivation for this work is toprovide a homological algebra characterization of PVMDs. To do so, we needthe notion of a w-flat module. Recently, modules of this type have receivedattention in several papers in the literature, see for example [3, 11, 21, 22].Throughout, R denotes a commutative ring with an identity element and allmodules are unitary.Now, we review some definitions and notation. Let J be an ideal of R.Following [25], J is called a Glaz-Vasconcelos ideal (a GV-ideal for short) if J isfinitely generated and the natural homomorphism ϕ : R → J

Let R be a prime ring with center [Formula: see text], δ: R → R a nonzero skew derivation, and n a fixed positive integer. In this paper, we show that R is a commutative ring if (i) [δ([x,y]),[x,y]]n = 0 for all x, y ∈ R or (ii) [Formula: see text] for all x ∈ R, except some specific cases.