Ring Morphism Research Articles

Centrally (quasi-)morphic modules

The main objective of this paper is investigating centrally (quasi-)morphic modules as a generalization of centrally morphic rings. We call an R-module M centrally quasi-morphic if for any f ∈ End R ( M ) , there exist central elements g , h ∈ End R ( M ) such that Ker f = Im g and Im f = Ker h . In addition, MR is said to be centrally morphic whenever g = h in the above definition. We show that for image-projective modules, these two notions coincide and every centrally quasi-morphic module is abelian. We prove that a module with strongly regular endomorphism ring (called strongly endoregular) is centrally morphic. Several properties of strongly endoregular modules are obtained.

Universally injective and integral contractions on relative Lipschitz saturation of algebras

In this work, we obtain contraction results for a class of diagrams of ring morphisms which strictly includes the ones obtained by Lipman. Further, we present some applications on quotient and in the changing of the base ring in the saturation.

On modules in which every finitely generated submodule is a kernel of an endomorphism

We study an R-module M in which every finitely generated submodule of M is a kernel of an endomorphism of M. Such modules are called Co-epi-finite-retractable (CEFR). We also consider CEFR condition on the co-local modules, submodules and factors of a CEFR module and direct sum of CEFR modules. Among other results, we prove that the injective hull of a simple module over a commutative Noetherian ring is uniserial if and only if it is CEFR.We investigate modules over a principal ideal ring, and show that all finitely generated torsion modules over a principal ideal domain are CEFR. Also, we show that every module over a commutative Köthe ring is CEFR. We also observe that a ring R is left pseudo morphic if and only if it is CEFR as a left R-module and we obtain some new properties of left pseudo morphic rings.

On (quasi-)morphic property of skew polynomial rings

The main objective of this paper is to study (quasi-)morphic property of skew polynomial rings. Let $R$ be a ring, $\sigma$ be a ring homomorphism on $R$ and $n\geq 1$. We show that $R$ inherits the quasi-morphic property from $R[x;\sigma]/(x^{n+1})$. It is also proved that the morphic property over $R[x;\sigma]/(x^{n+1})$ implies that $R$ is a regular ring. Moreover, we characterize a unit-regular ring $R$ via the morphic property of $R[x;\sigma]/(x^{n+1})$. We also investigate the relationship between strongly regular rings and centrally morphic rings. For instance, we show that for a domain $R$, $R[x;\sigma]/(x^{n+1})$ is (left) centrally morphic if and only if $R$ is a division ring and $\sigma(r)=u^{-1}ru$ for some $u\in R$. Examples which delimit and illustrate our results are provided.

Analytification, localization and homotopy epimorphisms

We study the interaction between various analytification functors, and a class of morphisms of rings, called homotopy epimorphisms. An analytification functor assigns to a simplicial commutative algebra over a ring R, along with a choice of Banach structure on R, a commutative monoid in the symmetric monoidal model category of simplicial ind-Banach R-modules. We show that several analytifications relevant to analytic geometry - such as Tate, overconvergent, Stein analytification, and formal completion - are homotopy epimorphisms. Another class of examples of homotopy epimorphisms arises from Weierstrass, Laurent and rational localizations in derived analytic geometry. As applications of this result, we prove that Hochschild homology and the cotangent complex are computable for analytic rings, and the computation relies only on known computations of Hochschild homology for polynomial rings. We show that in various senses, Hochschild homology as we define it commutes with localizations, analytifications and completions.

Structural results on lifting, orthogonality and finiteness of idempotents

In this paper, using the canonical correspondence between the idempotents and clopens, we obtain several new results on lifting idempotents. The Zariski clopens of the maximal spectrum are precisely determined, then as an application, lifting idempotents modulo the Jacobson radical is characterized. Lifting idempotents modulo an arbitrary ideal is also characterized in terms of certain connected sets related to that ideal. Then as an application, we obtain that the sum of a lifting ideal and a regular ideal is a lifting ideal. We prove that lifting idempotents preserves the orthogonality in countable cases. The lifting property of an arbitrary morphism of rings is characterized. As another major result, it is proved that the number of idempotents of a ring $R$ is finite if and only if it is of the form $2^{\kappa}$ where $\kappa$ is the cardinal of the connected components of Spec$(R)$. Finally, it is proved that the primitive idempotents of a zero dimensional ring are in 1-1 correspondence with the isolated points of its prime spectrum. These results either generalize or improve several important results in the literature.

Milnor excision for motivic spectra

Abstract We prove that the ∞ {\infty} -category of motivic spectra satisfies Milnor excision: if A → B {A\to B} is a morphism of commutative rings sending an ideal I ⊂ A {I\subset A} isomorphically onto an ideal of B, then a motivic spectrum over A is equivalent to a pair of motivic spectra over B and A / I {A/I} that are identified over B / I ⁢ B {B/IB} . Consequently, any cohomology theory represented by a motivic spectrum satisfies Milnor excision. We also prove Milnor excision for Ayoub’s étale motives over schemes of finite virtual cohomological dimension.

Stone type representations and dualities by power set ring

In this paper, it is shown that the Boolean ring of a commutative ring is isomorphic to the ring of clopens of its prime spectrum. In particular, Stone's Representation Theorem is generalized. The prime spectrum of the Boolean ring of a given ring R is identified with the Pierce spectrum of R . The discreteness of prime spectra is characterized. It is also proved that the space of connected components of a compact space X is isomorphic to the prime spectrum of the ring of clopens of X . As another major result, it is shown that a morphism of rings between complete Boolean rings preserves suprema if and only if the induced map between the corresponding prime spectra is an open map.

Some remarks about trunks and morphisms of neural codes

We give intrinsic characterizations of neural rings and homomorphisms between them. Also we introduce the notion of a basic monomial code map and characterize monomial code maps as compositions of basic monomial code maps. Finally, we characterize monomial isomorphisms between neural codes. Our work is based on the 2015 paper by C.~Curto and N.~Youngs about neural ring homomorphisms and maps between neural codes and on the 2018 paper by R.~Amzi Jeffs about morphisms of neural rings.

Classification of multivariate skew polynomial rings over finite fields via affine transformations of variables

In this work, free multivariate skew polynomial rings are considered, together with their quotients over ideals of skew polynomials that vanish at every point (which includes minimal multivariate skew polynomial rings). We provide a full classification of such multivariate skew polynomial rings (free or not) over finite fields. To that end, we first show that all ring morphisms from the field to the ring of square matrices are diagonalizable, and that the corresponding derivations are all inner derivations. Secondly, we show that all such multivariate skew polynomial rings over finite fields are isomorphic as algebras to a multivariate skew polynomial ring whose ring morphism from the field to the ring of square matrices is diagonal, and whose derivation is the zero derivation. Furthermore, we prove that two such representations only differ in a permutation on the field automorphisms appearing in the corresponding diagonal. The algebra isomorphisms are given by affine transformations of variables and preserve evaluations and degrees. In addition, ours proofs show that the simplified form of multivariate skew polynomial rings can be found computationally and explicitly.

Commutative morphic rings of stable range 2

It is know that a left quasi-morphic ring R is a ring of stable range 1 if and only if dim R = 0.
 In this paper it is shown that a commutative morphic ring R is a ring of stable range 2 if and only if dimR= 1.

EM-HERMITE RINGS

A ring $R$ is called EM-Hermite if for each $a,b\in R$, there exist $% a_{1},b_{1},d\in R$ such that $a=a_{1}d,b=b_{1}d$ and the ideal $% (a_{1},b_{1})$ is regular. We give several characterizations of EM-Hermite rings analogue to those for K-Hermite rings, for example, $R$ is an EM-Hermite ring if and only if any matrix in $M_{n,m}(R)$ can be written as a product of a lower triangular matrix and a regular $m\times m$ matrix. We relate EM-Hermite rings to Armendariz rings, rings with a.c. condition, rings with property A, EM-rings, generalized morphic rings, and PP-rings. We show that for an EM-Hermite ring, the polynomial ring and localizations are also EM-Hermite rings, and show that any regular row can be extended to regular matrix. We relate EM-Hermite rings to weakly semi-Steinitz rings, and characterize the case at which every finitely generated $R$-module with finite free resolution of length 1 is free.

RINGS WITH THE DUAL OF THE ISOMORPHISM THEOREM

A ring R satisfies the dual of the isomorphism theorem if R/Ra≅ 1(a) for all elements a of R, where 1(a) denotes the left annihilator. We call these rings left morphic. Examples include all unit regular rings and certain left uniserial local rings. We show that every left morphic ring is right principally injective, and use this to characterize the left perfect, right and left morphic rings.

Zariski locality of quasi-coherent sheaves associated with tilting

A classic result by Raynaud and Gruson says that the notion of an (infinite dimensional) vector bundle is Zariski local. This result may be viewed as a particular instance (for n = 0) of the locality of more general notions of quasi-coherent sheaves related to (infinite dimensional) n-tilting modules and classes. Here, we prove the latter locality for all n and all schemes. We also prove that the notion of a tilting module descends along arbitrary faithfully flat ring morphisms in several particular cases (including the case when the base ring is noetherian).

On a partially ordered set associated to ring morphisms

We associate to any ring R with identity a partially ordered set Hom(R), whose elements are all pairs (a,M), where a=ker⁡φ and M=φ−1(U(S)) for some ring morphism φ of R into an arbitrary ring S. Here U(S) denotes the group of units of S. The assignment R↦Hom(R) turns out to be a contravariant functor of the category Ring of associative rings with identity to the category ParOrd of partially ordered sets. The maximal elements of Hom(R) constitute a subset Max(R) which, for commutative rings R, can be identified with the Zariski spectrum Spec(R) of R. Every pair (a,M) in Hom(R) has a canonical representative, that is, there is a universal ring morphism ψ:R→S(R/a,M/a) corresponding to the pair (a,M), where the ring S(R/a,M/a) is constructed as a universal inverting R/a-ring in the sense of Cohn. Several properties of the sets Hom(R) and Max(R) are studied.

Change of rings and singularity categories

We investigate the behavior of singularity categories and stable categories of Gorenstein projective modules along a morphism of rings. The natural context to approach the problem is via change of rings, that is, the classical adjoint triple between the module categories. In particular, we identify conditions on the change of rings to induce functors between the two singularity categories or the two stable categories of Gorenstein projective modules. Moreover, we study this problem at the level of ‘big singularity categories’ in the sense of Krause [30]. Along the way we establish an explicit construction of a right adjoint functor between certain homotopy categories. This is achieved by introducing the notion of 0-cocompact objects in triangulated categories and proving a dual version of Bousfield's localization lemma. We provide applications and examples illustrating our main results.

Virtual rigid motives of semi-algebraic sets

Let k be a field of characteristic zero containing all roots of unity and $$K=k(( t))$$ . We build a ring morphism from the Grothendieck ring of semi-algebraic sets over K to the Grothendieck ring of motives of rigid analytic varieties over K. It extends the morphism sending the class of an algebraic variety over K to its cohomological motive with compact support. We show that it fits inside a commutative diagram involving Hrushovski and Kazhdan’s motivic integration and Ayoub’s equivalence between motives of rigid analytic varieties over K and quasi-unipotent motives over k; we also show that it satisfies a form of duality. This allows us to answer a question by Ayoub, Ivorra and Sebag about the analytic Milnor fiber.

Group-regular rings

We propose different generalizations of unit-regularity of elements in general rings (non necessarily unital rings). We then study general rings for which all elements have these properties. We notably compare them with unit-regular ideals and general rings with stable range one. We also prove that these rings are morphic rings.

Zariski Topology

Summary We formalize in the Mizar system [3], [4] basic definitions of commutative ring theory such as prime spectrum, nilradical, Jacobson radical, local ring, and semi-local ring [5], [6], then formalize proofs of some related theorems along with the first chapter of [1]. The article introduces the so-called Zariski topology. The set of all prime ideals of a commutative ring A is called the prime spectrum of A denoted by Spectrum A. A new functor Spec generates Zariski topology to make Spectrum A a topological space. A different role is given to Spec as a map from a ring morphism of commutative rings to that of topological spaces by the following manner: for a ring homomorphism h : A → B, we defined (Spec h) : Spec B → Spec A by (Spec h)(𝔭) = h −1(𝔭) where 𝔭 2 Spec B.

Some Extensions of Generalized Morphic Rings and EM-rings

Abstract Let R be a commutative ring with unity. The main objective of this article is to study the relationships between PP-rings, generalized morphic rings and EM-rings. Although PP-rings are included in the later rings, the converse is not in general true. We put necessary and sufficient conditions to ensure the converse using idealization and polynomial rings