Computing over the Reals (or an Arbitrary Ring)

1990 ◽  
pp. 25-26
Author(s):  
Lenore Blum
Keyword(s):  
Arbitrary Ring

scholarly journals Commutativity conditions on rings

1991 ◽  
Vol 44 (1) ◽  
pp. 45-47
Author(s):  
Paola Misso
Keyword(s):  
Positive Integer ◽  
Arbitrary Ring ◽  
Torsion Free

We prove the following result: let R be an arbitrary ring with centre Z such that for every x, y ∈ R, there exists a positive integer n = n(x, y) ≥ 1 such that (xy)n − ynxn ∈ Z and (yx)n − xnyn ∈ Z; then, if R has no non-zero nil ideals, R is commutative. We also prove a result on commutativity of general rings: if R is r!-torsion free and for all x, y ∈ R, [xr, ys] = 0 for fixed integers r ≥ s ≥ 1, then R is commutative. As a corollary we obtain that if R is (n + 1)!-torsion free and there exists a fixed n ≥ 1 such that (xy)n − ynxn = (yx)n − xnyn ∈ Z for all x, y ∈ R, then R is commutative.

scholarly journals Rings in which every element is either a sum or a difference of a nilpotent and an idempotent

2016 ◽  
Vol 15 (08) ◽  
pp. 1650148 ◽  
Author(s):  
Simion Breaz ◽  
Peter Danchev ◽  
Yiqiang Zhou
Keyword(s):  
Division Ring ◽  
Matrix Ring ◽  
Decomposition Theorem ◽  
Arbitrary Ring ◽  
Clean Rings ◽  
Clean Ring ◽  
Unital Rings ◽  
Nil Clean Ring ◽  
Comprehensive Study

Generalizing the notion of nil-cleanness from [A. J. Diesl, Nil clean rings, J. Algebra 383 (2013) 197–211], in parallel to [P. V. Danchev and W. Wm. McGovern, Commutative weakly nil clean unital rings, J. Algebra 425 (2015) 410–422], we define the concept of weak nil-cleanness for an arbitrary ring. Its comprehensive study in different ways is provided as well. A decomposition theorem of a weakly nil-clean ring is obtained. It is completely characterized when an abelian ring is weakly nil-clean. It is also completely determined when a matrix ring over a division ring is weakly nil-clean.

scholarly journals Group inverses and Drazin inverses of bidiagonal and triangular Toeqlitz matrices

1977 ◽  
Vol 24 (1) ◽  
pp. 10-34 ◽  
Author(s):  
R. E. Hartwig ◽  
J. Shoaf
Keyword(s):  
Toeplitz Matrices ◽  
Arbitrary Ring ◽  
Group Inverses ◽  
Necessary And Sufficient

AbstractNecessary and sufficient sonditions are given for the existence of the group and Drazin inverses of bidiagonal and triangular Toeplitz matrices over an arbitrary ring.

Rickart modules relative to singular submodule and dual Goldie torsion theory

2016 ◽  
Vol 15 (08) ◽  
pp. 1650142 ◽  
Author(s):  
Burcu Ungor ◽  
Sait Halicioglu ◽  
Abdullah Harmanci
Keyword(s):  
Torsion Theory ◽  
Arbitrary Ring ◽  
Rickart Modules

Let [Formula: see text] be an arbitrary ring with identity and [Formula: see text] a right [Formula: see text]-module with the ring [Formula: see text] End[Formula: see text] of endomorphisms of [Formula: see text]. The notion of an [Formula: see text]-inverse split module [Formula: see text], where [Formula: see text] is a fully invariant submodule of [Formula: see text], is defined and studied by the present authors. This concept produces Rickart submodules of modules in the sense of Lee, Rizvi and Roman. In this paper, we consider the submodule [Formula: see text] of [Formula: see text] as [Formula: see text] and [Formula: see text], and investigate some properties of [Formula: see text]-inverse split modules and [Formula: see text]-inverse split modules [Formula: see text]. Results are applied to characterize rings [Formula: see text] for which every free (projective) right [Formula: see text]-module [Formula: see text] is [Formula: see text]-inverse split for the preradicals such as [Formula: see text] and [Formula: see text].

The Maximal Co-Rational Extension by a Module

1966 ◽  
Vol 18 ◽  
pp. 953-962 ◽  
Author(s):  
R. C. Courter
Keyword(s):  
Projective Module ◽  
Unit Element ◽  
Projective Cover ◽  
Arbitrary Ring ◽  
Rational Extension ◽  
Image Position

Modules are S-modules where S is an arbitrary ring with or without a unit element. We consider a projective module P having a submodule K such that K + Y = P implies that the submodule Y is P (P, then, is a projective cover of P/K (Definition 4 in this section)) and we define the submodule X of P byOur main result states that up to isomorphism P/X is the maximal co-rational extension over P/K (by P/K, in the more precise wording of the title).

scholarly journals An equivalence induced by Ext and Tor applied to the finitistic weak dimension of coherent rings

1998 ◽  
Vol 40 (2) ◽  
pp. 167-176 ◽  
Author(s):  
Elwood Wilkins
Keyword(s):  
Representation Theory ◽  
Model Theory ◽  
Arbitrary Ring ◽  
Essential Component ◽  
Functor Categories ◽  
Weak Dimension ◽  
The 1960S ◽  
Almost Split Sequences ◽  
Coherent Rings

Let R be a ring, see below for other notation. The functor categories (mod-R, Ab) and ((R-mod)op, Ab) have received considerable attention since the 1960s. The first of these has achieved prominence in the model theory of modules and most particularly in the investigation of the representation theory of Artinian algebras. Both [11, Chapter 12] and [8] contain accounts of the use (mod-R, Ab) may be put to in the model theoretic setting, and Auslander's review, [1], details the application of (mod-R, Ab) to the study of Artinian algebras. The category ((R-mod)op, Ab) has been less fully exploited. Much work, however, has been devoted to the study of the transpose functor between R-mod and mod-R. Warfield's paper, [13], describes this for semiperfect rings, and this duality is an essential component in the construction of almost split sequences over Artinian algebras, see [4]. In comparison, the general case has been neglected. This paper seeks to remedy this situation, giving a concrete description of the resulting equivalence between (mod-R, Ab) and ((R-mod)op, Ab) for an arbitrary ring R.

FINITARY AUTOMORPHISM GROUPS OVER NON-COMMUTATIVE RINGS

2001 ◽  
Vol 64 (3) ◽  
pp. 611-623 ◽  
Author(s):  
B. A. F. WEHRFRITZ
Keyword(s):  
Division Ring ◽  
Automorphism Groups ◽  
Commutative Rings ◽  
Linear Groups ◽  
Locally Finite ◽  
Arbitrary Ring ◽  
Finitary Automorphism

The notion of a group of finitary automorphisms of an arbitrary module over an arbitrary ring is introduced, and it is shown how properties of such groups can be derived from the case where the ring is a division ring (that is, from the properties of finitary skew linear groups). The results are particularly strong if either the group is locally finite or the module is Noetherian.

scholarly journals Props in model categories and homotopy invariance of structures

2010 ◽  
Vol 17 (1) ◽  
pp. 79-160
Author(s):  
Benoit Fresse
Keyword(s):  
General Setting ◽  
Model Category ◽  
Topological Spaces ◽  
Algebra Structure ◽  
Model Structure ◽  
Homotopy Invariance ◽  
Model Categories ◽  
Arbitrary Ring ◽  
General Argument ◽  
Simplicial Sets

Abstract We prove that any category of props in a symmetric monoidal model category inherits a model structure. We devote an appendix, about half the size of the paper, to the proof of the model category axioms in a general setting. We need the general argument to address the case of props in topological spaces and dg-modules over an arbitrary ring, but we give a less technical proof which applies to the category of props in simplicial sets, simplicial modules, and dg-modules over a ring of characteristic 0. We apply the model structure of props to the homotopical study of algebras over a prop. Our goal is to prove that an object 𝑋 homotopy equivalent to an algebra 𝐴 over a cofibrant prop P inherits a P-algebra structure so that 𝑋 defines a model of 𝐴 in the homotopy category of P-algebras. In the differential graded context, this result leads to a generalization of Kadeishvili's minimal model of 𝐴∞-algebras.

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