Elementary equivalence of generalized incidence rings

In this paper, we prove that if two generalized incidence rings I(P ₁ ,R ₁) and I(P ₂ ,R ₂) are elementarily equivalent, then the corresponding ordered sets (P ₁ ,R ₁) and (P ₂ ,R ₂) are elementarily equivalent.     

Elementary equivalence of unitary linear groups over rings

We find necessary and sufficient conditions for two unitary linear groups over rings with involution with forms of hyperbolic rank at least 2 to be elementarily equivalent.     

Elementary equivalence of categories of modules over rings, endomorphism rings, and automorphism groups of modules

In this paper we give a brief review of some recent results of elementary equivalence of linear and algebraic groups and our last new results of elementary equivalence of categories of modules, endomorphism rings of modules, lattices of submodules of modules, and automorphism groups of modules. __________ Translated from Fundamentalnaya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 10, No. 2, pp. 51–134, 2004.     

Elementary equivalence of some rings of definable functions

We characterize elementary equivalences and inclusions between von Neumann regular real closed rings in terms of their boolean algebras of idempotents, and prove that their theories are always decidable. We then show that, under some hypotheses, the map sending an L-structure R to the L-structure of definable functions from R ⁿ to R preserves elementary inclusions and equivalences and gives a structure with a decidable theory whenever R is decidable. We briefly consider structures of definable functions satisfying an extra condition such as continuity.      

Elementary equivalence of endomorphism rings of Abelian p-groups

In this paper we study a relationship between elementary equivalence of endomorphism rings of Abelian p-groups and second-order equivalence of the corresponding Abelian p-groups. __________ Translated from Fundamentalnaya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 10, No. 2, pp. 135–224, 2004.     

Elementary equivalence of semigroups of invertible matrices with nonnegative elements over commutative partially ordered rings

In the paper, we prove that if two semigroups of invertible matrices with nonnegative elements over partially ordered commutative rings are elementarily equivalent, then their dimensions coincide and the corresponding semirings of nonnegative elements are elementarily equivalent.     

Elementary equivalence of rings with finitely generated additive groups

We give algebraic characterizations of elementary equivalence between rings with finitely generated additive groups. They are similar to those previously obtained for finitely generated nilpotent groups. Here, the rings are not supposed associative, commutative or unitary.     

Elementary equivalence of Chevalley groups over fields

It is proved that (elementary) Chevalley groups G _π(Φ,K) and G _π′(Φ′,K′) (or E _π(Φ,K) and E _π′(Φ′,K′)) over infinite fields K and K′ of characteristic different from 2, with weight lattices Λ and Λ′, respectively, are elementarily equivalent if and only if the root systems Φ and Φ′ are isomorphic, the fields K and K′ are elementarily equivalent, and the lattices Λ and Λ′ coincide. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 8, pp. 29–77, 2006.     

Elementary matrix reduction over J-stable rings

A commutative ring R is J-stable provided that R∕aR has stable range 1 for all a?J(R). A commutative ring R in which every finitely generated ideal principal is called a Bézout ring. A ring R is an elementary divisor ring provided that every matrix over R admits a diagonal reduction. We prove that a J-stable ring is a Bézout ring if and only if it is an elementary divisor ring. Further, we prove that every J-stable ring is strongly completable. Various types of J-stable rings are provided. Many known results are thereby generalized to much wider class of rings, e.g. [3 Gillman, L., Henriksen, M. (1956). Some remarks about elementary divisor rings. Trans. Amer. Math. Soc. 82:362–365.[Crossref] , [Google Scholar], Theorem 8], [4 Larsen, M., Lewis, W., Shores, T. (1974). Elementary divisor rings and finitely presented modules. Trans. Amer. Math. Soc. 187:231–248.[Crossref], [Web of Science ®] , [Google Scholar], Theorem 4.1], [7 McGovern, W. W. (2008). Bézout rings with almost stable range 1. J. Pure Appl. Algebra 212:340–348.[Crossref], [Web of Science ®] , [Google Scholar], Theorem 3.7], [8 Moore, M. E. (1975). A strongly complement property of Dedekind domain. Czechoslovak Math. J. 25(100):282–283. [Google Scholar], Theorem], [9 Moore, M., Steger, A. (1971). Some results on completability in commutative rings. Pacific J. Math. 37:453–460.[Crossref], [Web of Science ®] , [Google Scholar], Theorem 2.1], [14 Zabavsky, B. V. (1996). Generalized adequate rings. Ukrainian Math. J. 48:614–617.[Crossref] , [Google Scholar], Theorem 1] and [18 Zabavsky, B. V., Komarnyts’kyi, M. Y. (2010). Cohn-type theorem for adequacy and elementary divisor rings. J. Math. Sci. 167:107–111.[Crossref] , [Google Scholar], Theorem 7].     

Elementary matrix reduction over certain rings

We explore elementary matrix reduction over certain rings characterized by properties related to stable range. Let R be a commutative ring. We call R locally stable if aR+bR = R??x∈R such that R∕(a+bx)R has stable range 1. We study locally stable rings and prove that every locally stable Bézout ring is an elementary divisor ring. Many known results on domains are thereby generalized.     

Induced liftings, exchange rings and semi-perfect algebras

Several important classes of rings can be characterized in terms of liftings of idempotents with respect to various ideals: classical examples are semi-perfect rings, semi-regular rings and exchange rings. We begin with a study of some extensions of the concept of idempotent lifting and prove the generalizations of some classical lifting theorems. Then we describe the method of induced liftings, which allows us to transfer liftings from a ring to its subrings. Using this method we are able to show that under certain assumptions a subring of an exchange ring is also an exchange ring, and to prove that a finite algebra over a commutative local ring is semi-perfect, provided it can be suitably represented in an exchange ring.     

On the equivalence of codes over rings and modules

In light of the result by Wood that codes over every finite Frobenius ring satisfy a version of the MacWilliams equivalence theorem, a proof for the converse is considered. A strategy is proposed that would reduce the question to problems dealing only with matrices over finite fields. Using this strategy, it is shown, among other things, that any left MacWilliams basic ring is Frobenius. The results and techniques in the paper also apply to related problems dealing with codes over modules.     

Characteristics of invariant weights related to code equivalence over rings

The Equivalence Theorem states that, for a given weight on an alphabet, every isometry between linear codes extends to a monomial transformation of the entire space. This theorem has been proved for several weights and alphabets, including the original MacWilliams’ Equivalence Theorem for the Hamming weight on codes over finite fields. The question remains: What conditions must a weight satisfy so that the Extension Theorem will hold? In this paper we provide an algebraic framework for determining such conditions, generalising the approach taken in Greferath and Honold (Proceedings of the Tenth International Workshop in Algebraic and Combinatorial Coding Theory (ACCT-10), pp. 106–111. Zvenigorod, Russia, 2006).     

Elementary regular rings

Translated fromAlgebra i Logika, Vol. 32, No. 4, pp. 387–401, July–August, 1993.     

The complexity of the equivalence and equation solvability problems over nilpotent rings and groups

It is proved that the equation solvability problem can be solved in polynomial time for finite nilpotent rings. Ramsey’s theorem is employed in the proof. Then, using the same technique, a theorem of Goldmann and Russell is reproved: the equation solvability problem can be solved in polynomial time for finite nilpotent groups.