Contramodules over pro-perfect topological rings

For four wide classes of topological rings R \mathfrak{R} , we show that all flat left R \mathfrak{R} -contramodules have projective covers if and only if all flat left R \mathfrak{R} -contramodules are projective if and only if all left R \mathfrak{R} -contramodules have projective covers if and only if all descending chains of cyclic discrete right R \mathfrak{R} -modules terminate if and only if all the discrete quotient rings of R \mathfrak{R} are left perfect. Three classes of topological rings for which this holds are the complete, separated topological associative rings with a base of neighborhoods of zero formed by open two-sided ideals such that either the ring is commutative, or it has a countable base of neighborhoods of zero, or it has only a finite number of semisimple discrete quotient rings. The fourth class consists of all the topological rings with a base of neighborhoods of zero formed by open right ideals which have a closed two-sided ideal with certain properties such that the quotient ring is a topological product of rings from the previous three classes. The key technique on which the proofs are based is the contramodule Nakayama lemma for topologically T-nilpotent ideals.

A Certain Structure of Bipolar Fuzzy Subrings

The role of symmetry in ring theory is universally recognized. The most directly definable universal relation in a symmetric set theory is isomorphism. This article develops a certain structure of bipolar fuzzy subrings, including bipolar fuzzy quotient ring, bipolar fuzzy ring homomorphism, and bipolar fuzzy ring isomorphism. We define (α,β)-cut of bipolar fuzzy set and investigate the algebraic attributions of this phenomenon. We also define the support set of bipolar fuzzy set and prove various important properties relating to this concept. Additionally, we define bipolar fuzzy homomorphism by using the notion of natural ring homomorphism. We also establish a bipolar fuzzy homomorphism between bipolar fuzzy subring of the quotient ring and bipolar fuzzy subring of this ring. We constituted a significant relationship between two bipolar fuzzy subrings of quotient rings under a given bipolar fuzzy surjective homomorphism. We present the construction of an induced bipolar fuzzy isomorphism between two related bipolar fuzzy subrings. Moreover, to discuss the symmetry between two bipolar fuzzy subrings, we present three fundamental theorems of bipolar fuzzy isomorphism.

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.

Truncations of ordered abelian groups

We give axioms for a class of ordered structures, called truncated ordered abelian groups (TOAG’s) carrying an addition. TOAG’s come naturally from ordered abelian groups with a 0 and a $$+$$ + , but the addition of a TOAG is not necessarily even a cancellative semigroup. The main examples are initial segments $$[0, \tau ]$$ [ 0 , τ ] of an ordered abelian group, with a truncation of the addition. We prove that any model of these axioms (i.e. a truncated ordered abelian group) is an initial segment of an ordered abelian group. We define Presburger TOAG’s, and give a criterion for a TOAG to be a Presburger TOAG, and for two Presburger TOAG’s to be elementarily equivalent, proving analogues of classical results on Presburger arithmetic. Their main interest for us comes from the model theory of certain local rings which are quotients of valuation rings valued in a truncation [0, a] of the ordered group $${\mathbb {Z}}$$ Z or more general ordered abelian groups, via a study of these truncations without reference to the ambient ordered abelian group. The results are used essentially in a forthcoming paper (D’Aquino and Macintyre, The model theory of residue rings of models of Peano Arithmetic: The prime power case, 2021, arXiv:2102.00295) in the solution of a problem of Zilber about the logical complexity of quotient rings, by principal ideals, of nonstandard models of Peano arithmetic.

Hv-Field of Fractions and Hv-Quotient Rings

A larger class of algebraic hyperstructures satisfying the ring (field)-likeaxioms is the class of Hv-rings (Hv-fields). In this paper, we define the Hv-integraldomain and introduce the Hv-field of fractions of an Hv-integral domain. Also, theHv-quotient ring and some relative theorems are presented. Finally, some interestingresults about the Hv-rings of fractions, Hv-quotient rings and the relations betweenthem are proved.

Covering classes and 1-tilting cotorsion pairs over commutative rings

We are interested in characterising the commutative rings for which a 1-tilting cotorsion pair ( 𝒜 , 𝒯 ) {(\mathcal{A},\mathcal{T})} provides for covers, that is when the class 𝒜 {\mathcal{A}} is a covering class. We use Hrbek’s bijective correspondence between the 1-tilting cotorsion pairs over a commutative ring R and the faithful finitely generated Gabriel topologies on R. Moreover, we use results of Bazzoni–Positselski, in particular a generalisation of Matlis equivalence and their characterisation of covering classes for 1-tilting cotorsion pairs arising from flat injective ring epimorphisms. Explicitly, if 𝒢 {\mathcal{G}} is the Gabriel topology associated to the 1-tilting cotorsion pair ( 𝒜 , 𝒯 ) {(\mathcal{A},\mathcal{T})} , and R 𝒢 {R_{\mathcal{G}}} is the ring of quotients with respect to 𝒢 {\mathcal{G}} , we show that if 𝒜 {\mathcal{A}} is covering, then 𝒢 {\mathcal{G}} is a perfect localisation (in Stenström’s sense [B. Stenström, Rings of Quotients, Springer, New York, 1975]) and the localisation R 𝒢 {R_{\mathcal{G}}} has projective dimension at most one as an R-module. Moreover, we show that 𝒜 {\mathcal{A}} is covering if and only if both the localisation R 𝒢 {R_{\mathcal{G}}} and the quotient rings R / J {R/J} are perfect rings for every J ∈ 𝒢 {J\in\mathcal{G}} . Rings satisfying the latter two conditions are called 𝒢 {\mathcal{G}} -almost perfect.