Classical Quotient Ring Research Articles

Characterization of rings whose classical quotient rings are perfect rings

Characterization of rings whose classical quotient rings are perfect rings

On some properties of the space of minimal prime ideals of 𝐶𝑐 (𝑋)

In this article we consider some relations between the topological properties of the spaces X and Min(Cc (X)) with algebraic properties of Cc (X). We observe that the compactness of Min(Cc (X)) is equivalent to the von-Neumann regularity of qc (X), the classical ring of quotients of Cc (X). Furthermore, we show that if 𝑋 is a strongly zero-dimensional space, then each contraction of a minimal prime ideal of 𝐶(𝑋) is a minimal prime ideal of Cc(X) and in this case 𝑀𝑖𝑛(𝐶(𝑋)) and Min(Cc (X)) are homeomorphic spaces. We also observe that if 𝑋 is an Fc-space, then Min(Cc (X)) is compact if and only if 𝑋 is countably basically disconnected if and only if Min(Cc(X)) is homeomorphic with β0X. Finally, by introducing zoc-ideals, countably cozero complemented spaces, we obtain some conditions on X for which Min(Cc (X)) becomes compact, basically disconnected and extremally disconnected.

Stable range conditions for abelian and duo rings

The article deals with the following question: when does the classical ring of quotientsof a duo ring exist and idempotents in the classical ring of quotients $Q_{Cl} (R)$ are thereidempotents in $R$? In the article we introduce the concepts of a ring of (von Neumann) regularrange 1, a ring of semihereditary range 1, a ring of regular range 1. We find relationshipsbetween the introduced classes of rings and known ones for abelian and duo rings.We proved that semihereditary local duo ring is a ring of semihereditary range 1. Also it was proved that a regular local Bezout duo ring is a ring of stable range 2. In particular, the following Theorem 1 is proved: For an abelian ring $R$ the following conditions are equivalent:$1.$\ $R$ is a ring of stable range 1; $2.$\ $R$ is a ring of von Neumann regular range 1.
 The paper also introduces the concept of the Gelfand element and a ring of the Gelfand range 1 for the case of a duo ring. Weproved that the Hermite duo ring of the Gelfand range 1 is an elementary divisor ring (Theorem 3).

Characterizations of maximal z°-ideals of C(X) and real maximal ideals of q(X)

In this paper we first give elementwise characterizations of real maximal ideals and maximal z°-ideals of C(X). Next, using this we characterize the real maximal ideals of the classical ring of quotients q(X) of C(X) and also maximal z°-ideals of factor rings of C(X). We show that every real maximal ideal of q(X) is precisely the extension of a real maximal ideal of C(X) which is also a z° -ideal, i.e., the extension of Mp for some almost P -point p of υX. Using this fact it turns out that in contrast to C(X), q(X) may not contain any real maximal ideal. We observe that every maximal ideal of q(X) is real if and only if X is a pseudocompact almost P -space, i.e., q(X) = C*(X). We also observe that the real maximal ideals of q(X) are precisely the extensions of the real maximal ideals of C(X) when and only when X is an almost P -space, i.e., q(X) = C(X).

On Fundamental Theorems of Fuzzy Isomorphism of Fuzzy Subrings over a Certain Algebraic Product

In this study, we define the concept of an ω-fuzzy set ω-fuzzy subring and show that the intersection of two ω-fuzzy subrings is also an ω-fuzzy subring of a given ring. Moreover, we give the notion of an ω-fuzzy ideal and investigate different fundamental results of this phenomenon. We extend this ideology to propose the notion of an ω-fuzzy coset and develop a quotient ring with respect to this particular fuzzy ideal analog into a classical quotient ring. Additionally, we found an ω-fuzzy quotient subring. We also define the idea of a support set of an ω-fuzzy set and prove various important characteristics of this phenomenon. Further, we describe ω-fuzzy homomorphism and ω-fuzzy isomorphism. We establish an ω-fuzzy homomorphism between an ω-fuzzy subring of the quotient ring and an ω-fuzzy subring of this ring. We constitute a significant relationship between two ω-fuzzy subrings of quotient rings under the given ω-fuzzy surjective homomorphism and prove some more fundamental theorems of ω-fuzzy homomorphism for these specific fuzzy subrings. Finally, we present three fundamental theorems of ω-fuzzy isomorphism.

The rings where zero-divisor polynomials have zero-divisor coefficients

Based on McCoy’s theorem on commutative rings, Nielsen called a ring R right McCoy if the equation f(x)g(x)=0 implies f(x)c=0 for some nonzero element c in R, where f(x) and g(x) are nonzero polynomials in R[x]. In this paper, a class of rings is introduced and called ZPZC rings, containing McCoy rings, and then their properties are investigated. Also, associations are found between ZPZC rings and other related rings. Moreover, several extensions of ZPZC rings are studied, including matrix rings, trivial extensions, Hochschild extensions and classical quotient rings.

$\mathcal P_1$-covers over commutative rings

In this paper we consider the class \mathcal P_1(R) of modules of projective dimension at most one over a commutative ring R and we investigate when \mathcal P_1(R) is a covering class. More precisely, we investigate Enochs' Conjecture, that is the question of whether \mathcal P_1(R) is covering necessarily implies that \mathcal P_1(R) is closed under direct limits. We answer the question affirmatively in the case of a commutative semihereditary ring R . This gives an example of a cotorsion pair (\mathcal P_1(R), \mathcal P_1(R)^\perp) which is not necessarily of finite type such that \mathcal P_1(R) satisfies Enochs' Conjecture. Moreover, we describe the class \varinjlim \mathcal P_1(R) over (not necessarily commutative) rings which admit a classical ring of quotients.

C(X): Something old and something new

The aim of this paper is twofold. Firstly, to give short proofs of the celebrated results of Rudd (1970), De Marco (1972), Brookshear (1977), Neville (1990), and Vechtomov (1996) with the help of some well-known results in ring theory. Secondly, to establish new algebraic characterizations for when C(X) is von Neumann regular, semihereditary, or a Bézout ring. The notions of the greatest common divisor and least common multiple in the ring C(X) are studied. Finally, the space X, for which every idempotent of the classical quotient ring of C(X) is actually an element of C(X) itself, is completely determined.

Flatness of Noetherian Hopf algebras over coideal subalgebras

It is proved in the paper that a Noetherian residually finite-dimensional Hopf algebra H is a flat module over any right Noetherian right coideal subalgebra A. In the case when A is a Hopf subalgebra we get faithful flatness. These results are obtained by verifying the existence of classical quotient rings of A and H. It is also proved that the antipode of either right or left Noetherian residually finite-dimensional Hopf algebra is bijective. As a consequence, such a Hopf algebra is right and left Noetherian simultaneously.

Criteria for the Validity of Goldie’s Theorems for Graded Rings

We specify conditions on a group G that are necessary and sufficient for analogs of Goldie’s theorems to hold in a class of G-graded rings, i.e., for every G-graded gr-prime (gr-semiprime) right Goldie ring to possess a completely gr-reducible graded classical right ring of quotients.

Equivalence of some homological conditions for ring epimorphisms

Let R⊆Q be rings such that the inclusion φ:R→Q is an epimorphism in the category of rings and Q is flat both as a right and as a left R-module. Assume that F.dim(QQ)=0, that is, that every left Q-module has projective dimension 0 or ∞. We prove that the following conditions are equivalent: (i) Flat left R-modules are strongly flat. (ii) Matlis-cotorsion left R-modules are Enochs-cotorsion. (iii)h-divisible left R-modules are weak-injective. (iv) Homomorphic images of weak-injective left R-modules are weak-injective. (v) Homomorphic images of injective left R-modules are weak-injective. (vi) Left R-modules of weak dimension ≤1 are of projective dimension ≤1. (vii) The cotorsion pairs (P1,HD) and (F1,WI) coincide. Moreover, if R is a right and left Ore ring, S is its set of regular elements and Q=R[S−1]=[S−1]R is the classical ring of quotients of R, then the seven conditions above are also equivalent to: (viii) Divisible left R-modules are weak-injective. This extends a result by Fuchs and Salce (2018) [10] for modules over a commutative ring R.

Homomorphisms from $$C(X,\mathbb {Z})$$ C ( X , Z ) into a ring of continuous functions

Let X be a zero-dimensional space and Y be a Tychonoff space. We show that every non-zero ring homomorphism $$\Phi :C(X,\mathbb {Z})\rightarrow C(Y)$$ can be induced by a continuous function $$\pi :Y\rightarrow \upsilon _0X.$$ Using this, it turns out that the kernel of such homomorphisms is equal to the intersection of some family of minimal prime ideals in $${{\mathrm{MinMax}}}\left( C(X,\mathbb {Z})\right) .$$ As a consequence, we are able to obtain the fact that the factor ring $$\frac{C(X,\mathbb {Z})}{C_F(X,\mathbb {Z})}$$ is a subring of some ring of continuous functions if and only if each infinite subset of isolated points of X has a limit point in $$\upsilon _0X.$$ This implies that for an arbitrary infinite set X, the factor ring $$\frac{\prod _{_{x\in X}}\mathbb {Z}_{_{x}}}{\oplus _{_{x\in X}}\mathbb {Z}_{_{x}}}$$ is not embedded in any ring of continuous functions. The classical ring of quotients of the factor ring $$\frac{C(X,\mathbb {Z})}{C_F(X,\mathbb {Z})}$$ is fully characterized. Finally, it is shown that the factor ring $$\frac{C(X,\mathbb {Z})}{C_F(X,\mathbb {Z})}$$ is an I-ring if and only if each infinite subset of isolated points on X has a limit point in $$\upsilon _0X$$ and $$\upsilon _0X{\setminus }\mathbb {I}(X)$$ is an extremally disconnected $$C_{\mathbb {Z}}$$ -subspace of $$\upsilon _0X,$$ where $$\mathbb {I}(X)$$ is the set of all isolated points of X.

Strongly _-clean properties and rings of functions

ARis the sum of a unit and a projection that commute with each other. Inthis paper, we explore strong -cleanness of rings of continuous functions overspectrum spaces. We prove that a -ring R is strongly -clean if and only if Ris an abelian exchange ring and C(X) C (X) is-clean, if and only if R isan abelian exchange ring and the classical ring of quotients q(C(X)) of C(X)is -clean, where X is a spectrum space of R

ON MODULES OVER COMMUTATIVE RINGS

Our main purpose is to extend several results of interest that have been proved for modules over integral domains to modules over arbitrary commutative rings $R$ with identity. The classical ring of quotients $Q$ of $R$ will play the role of the field of quotients when zero-divisors are present. After discussing torsion-freeness and divisibility (Sections 2–3), we study Matlis-cotorsion modules and their roles in two category equivalences (Sections 4–5). These equivalences are established via the same functors as in the domain case, but instead of injective direct sums $\oplus Q$ one has to take the full subcategory of $Q$-modules into consideration. Finally, we prove results on Matlis rings, i.e. on rings for which $Q$ has projective dimension $1$ (Theorem 6.4).

Commutative rings in which zero-components of essential primes are essential

For a prime ideal [Formula: see text] of a commutative ring [Formula: see text] with identity, we denote (as usual) by [Formula: see text] its zero-component; that is, the set of members of [Formula: see text] that are annihilated by nonmembers of [Formula: see text]. We study rings in which [Formula: see text] is an essential ideal, whenever [Formula: see text] is an essential prime ideal. We characterize them in terms of the lattices (which are, in fact, complete Heyting algebras) of their radical ideals. We prove that the classical ring of quotients of any ring of this kind is itself of this kind. We show that direct products of rings of this kind are themselves of this kind. We observe that the ring of real-valued continuous functions on a Tychonoff space is of this kind precisely when the underlying set of the space is infinite. Replacing [Formula: see text] with the pure part of [Formula: see text], we obtain a formally stronger variant which is still characterizable in terms of the lattices of radical ideals.

On rings with annihilator condition

In this paper we study rings R with the property that every finitely generated ideal of R consisting entirely of zero divisors has a nonzero annihilator. The class of commutative rings with this property is quite large; for example, noetherian rings, rings whose prime ideals are maximal, the polynomial ring R[x] and rings whose classical ring of quotients are von Neumann regular. We continue to study conditions under which right mininjective rings, right FP-injective rings, right weakly continuous rings, right extending rings, one sided duo rings, semiregular rings and semiperfect rings have this property.

Inglés

We consider a first approach to the notion of Armendariz ring for a skew Poincare-Birkho-Witt (PBW for short) extension, and its lassi al ring of quotients. As an immediate application of this treatment, we study the properties Baer, quasi-Baer, p.p. and p.q.-Baer rings for these extensions. In this way, we generalize several results in the literature concerning Ore extensions and skew PBW extensions. To ite this article: A. Reyes, H. Suarez, Armendariz property for skew PBW extensions and their classical ring of quotients, Rev. Integr. Temas Mat. 34 (2016), No. 2, 147-168.

Ideal structure of the classical ring of quotients of C(X)

In this paper we show that the set of z-ideals and the set of z∘ ideals (=d-ideals) of the classical ring of quotients q(R) (q(X)) of a reduced ring R with property A (C(X)) coincide. Using this fact, we observe that each maximal ideal of q(R) is the extension of a maximal z∘-ideal of R. The members of maximal z∘-ideals of C(X) contained in a given maximal ideal are topologically characterized and using this, it turns out that the extension Op of each Op, p∈βX is a maximal ideal of q(X) if and only if X is a basically disconnected space. Topological spaces X are also characterized for which every Op is contained in a unique maximal ideal of q(X) and in this case, the maximal ideals of q(X) are characterized. Finally, using the concept of z-ideal in q(X), we characterize the regularity of q(X). For instance, we observe that q(X) is regular if and only if for each f∈C(X), there exists a regular (non-zero divisor) element r such that Z(f)∩cozr is open in cozr or equivalently, |f||r| is an idempotent in q(X).

Invariants and rings of quotients of H-semiprime H-module algebras satisfying a polynomial identity

We consider an action of a finite-dimensional Hopf algebra H on a PI-algebra. We prove that an H-semiprime H-module algebra A has a Frobenius artinian classical ring of quotients Q, provided that A has a finite set of H-prime ideals with zero intersection. The ring of quotients Q is an H-semisimple H-module algebra and a finitely generated module over the subalgebra of central invariants. Moreover, if algebra A is a projective module of constant rank over its center, then A is integral over its subalgebra of central invariants.

Lamplighter groups and von Neumann‘s continuous regular ring

Let Γ \Gamma be a discrete group. Following Linnell and Schick one can define a continuous ring c ( Γ ) c(\Gamma ) associated with Γ \Gamma . They proved that if the Atiyah Conjecture holds for a torsion-free group Γ \Gamma , then c ( Γ ) c(\Gamma ) is a skew field. Also, if Γ \Gamma has torsion and the Strong Atiyah Conjecture holds for Γ \Gamma , then c ( Γ ) c(\Gamma ) is a matrix ring over a skew field. The simplest example when the Strong Atiyah Conjecture fails is the lamplighter group Γ = Z 2 ≀ Z \Gamma =\mathbb {Z}_2\wr \mathbb {Z} . It is known that C ( Z 2 ≀ Z ) \mathbb {C}(\mathbb {Z}_2\wr \mathbb {Z}) does not even have a classical ring of quotients. Our main result is that if H H is amenable, then c ( Z 2 ≀ H ) c(\mathbb {Z}_2\wr H) is isomorphic to a continuous ring constructed by John von Neumann in the 1930s.