scholarly journals ABSENCE OF TORSION FOR NK1(R) OVER ASSOCIATIVE RINGS

2011 ◽  
Vol 10 (04) ◽  
pp. 793-799 ◽  
Author(s):  
RABEYA BASU
Keyword(s):  
Commutative Ring ◽  
Associative Ring ◽  
Module Structure ◽  
Lecture Notes ◽  
Associative Rings

When R is a commutative ring with identity, and if k ∈ ℕ, with kR = R, then it was shown in [C. Weibel, Mayer–Vietoris Sequence and Module Structure on NK0, Lecture Notes in Mathematics, Vol. 854 (Springer, 1981), pp. 466–498] that SK 1(R[X]) has no k-torsion. We prove this result for any associative ring R with identity in which kR = R.

Modules with annihilation property

2020 ◽  
pp. 2150126
Author(s):  
Rasul Mohammadi ◽  
Ahmad Moussavi ◽  
Masoome Zahiri
Keyword(s):  
Commutative Ring ◽  
Associative Ring ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Finitely Generated ◽  
Ideal Formula ◽  
Ordered Monoid ◽  
Zero Divisors ◽  
The Right ◽  
Totally Ordered Monoid

Let [Formula: see text] be an associative ring with identity. A right [Formula: see text]-module [Formula: see text] is said to have Property ([Formula: see text]), if each finitely generated ideal [Formula: see text] has a nonzero annihilator in [Formula: see text]. Evans [Zero divisors in Noetherian-like rings, Trans. Amer. Math. Soc. 155(2) (1971) 505–512.] proved that, over a commutative ring, zero-divisor modules have Property ([Formula: see text]). We study and construct various classes of modules with Property ([Formula: see text]). Following Anderson and Chun [McCoy modules and related modules over commutative rings, Comm. Algebra 45(6) (2017) 2593–2601.], we introduce [Formula: see text]-dual McCoy modules and show that, for every strictly totally ordered monoid [Formula: see text], faithful symmetric modules are [Formula: see text]-dual McCoy. We then use this notion to give a characterization for modules with Property ([Formula: see text]). For a faithful symmetric right [Formula: see text]-module [Formula: see text] and a strictly totally ordered monoid [Formula: see text], it is proved that the right [Formula: see text]-module [Formula: see text] is primal if and only if [Formula: see text] is primal with Property ([Formula: see text]).

scholarly journals HYPERCOMMUTING VALUES IN ASSOCIATIVE RINGS WITH UNITY

2013 ◽  
Vol 94 (2) ◽  
pp. 181-188
Author(s):  
VINCENZO DE FILIPPIS ◽  
GIOVANNI SCUDO
Keyword(s):  
Commutative Ring ◽  
Polynomial Identity ◽  
Multilinear Polynomial ◽  
Associative Rings

AbstractLet $K$ be a commutative ring with unity, $R$ an associative $K$-algebra of characteristic different from $2$ with unity element and no nonzero nil right ideal, and $f({x}_{1} , \ldots , {x}_{n} )$ a multilinear polynomial over $K$. Assume that, for all $x\in R$ and for all ${r}_{1} , \ldots , {r}_{n} \in R$ there exist integers $m= m(x, {r}_{1} , \ldots , {r}_{n} )\geq 1$ and $k= k(x, {r}_{1} , \ldots , {r}_{n} )\geq 1$ such that $\mathop{[{x}^{m} , f({r}_{1} , \ldots , {r}_{n} )] }\nolimits_{k} = 0$. We prove that: (1) if $\text{char} (R)= 0$ then $f({x}_{1} , \ldots , {x}_{n} )$ is central-valued on $R$; and (2) if $\text{char} (R)= p\gt 2$ and $f({x}_{1} , \ldots , {x}_{n} )$ is not a polynomial identity in $p\times p$ matrices of characteristic $p$, then $R$ satisfies ${s}_{n+ 2} ({x}_{1} , \ldots , {x}_{n+ 2} )$ and for any ${r}_{1} , \ldots , {r}_{n} \in R$ there exists $t= t({r}_{1} , \ldots , {r}_{n} )\geq 1$ such that ${f}^{{p}^{t} } ({r}_{1} , \ldots , {r}_{n} )\in Z(R)$, the center of $R$.

Zariski-like topologies for lattices with applications to modules over associative rings

2019 ◽  
Vol 18 (07) ◽  
pp. 1950131
Author(s):  
Jawad Abuhlail ◽  
Hamza Hroub
Keyword(s):  
Complete Lattice ◽  
Associative Ring ◽  
Sufficient Conditions ◽  
Proper Class ◽  
Topological Properties ◽  
Left Module ◽  
Separation Axioms ◽  
Prime Submodules ◽  
Algebraic Properties ◽  
Associative Rings

We study Zariski-like topologies on a proper class [Formula: see text] of a complete lattice [Formula: see text]. We consider [Formula: see text] with the so-called classical Zariski topology [Formula: see text] and study its topological properties (e.g. the separation axioms, the connectedness, the compactness) and provide sufficient conditions for it to be spectral. We say that [Formula: see text] is [Formula: see text]-top if [Formula: see text] is a topology. We study the interplay between the algebraic properties of an [Formula: see text]-top complete lattice [Formula: see text] and the topological properties of [Formula: see text] Our results are applied to several spectra which are proper classes of [Formula: see text] where [Formula: see text] is a nonzero left module over an arbitrary associative ring [Formula: see text] (e.g. the spectra of prime, coprime, fully prime submodules) of [Formula: see text] as well as to several spectra of the dual complete lattice [Formula: see text] (e.g. the spectra of first, second and fully coprime submodules of [Formula: see text]).

scholarly journals Weakly prime left ideals in general rings

1985 ◽  
Vol 32 (3) ◽  
pp. 357-360
Author(s):  
Halina France-Jackson
Keyword(s):  
Left Ideal ◽  
Associative Ring ◽  
Almost All ◽  
Associative Rings

A.P.J. van der Walt introduced the concept of a weakly prime left ideal of an associative ring with unity. It is the purpose of the present paper to extend to general, that is not necessarily with unity associative rings, this concept as well as almost all results of van der Walt for rings with unity.

Commutators in associative rings

1953 ◽  
Vol 49 (4) ◽  
pp. 590-594 ◽  
Author(s):  
M. P. Drazin ◽  
K. W. Gruenberg
Keyword(s):  
Associative Ring ◽  
Lie Ring ◽  
Addition And Subtraction ◽  
Associative Rings

Let R be an arbitrary associative ring, and X a set of generators of R. The elements of X generate a Lie ring, [X], say, with respect to the addition and subtraction in R, and the multiplication [a, b] = ab − ba. In this note we shall be concerned with the following question: if [X] is given to be nilpotent as a Lie ring, what does this imply about R?

scholarly journals C-Commutativity

1980 ◽  
Vol 30 (2) ◽  
pp. 252-255 ◽  
Author(s):  
T. Cheatham ◽  
E. Enochs
Keyword(s):  
Finite Number ◽  
Commutative Ring ◽  
Associative Ring ◽  
Nonzero Divisor ◽  
Special Case

AbstractAn associative ring R with identity is said to be c-commutative for c ∈ R if a, b ∈ R and ab = c implies ba = c. Taft has shown that if R is c-commutative where c is a central nonzero divisor]can be omitted. We show that in R[x] is h(x)-commutative for any h(x) ∈ R [x] then so is R with any finite number of (commuting) indeterminates adjoined. Examples adjoined. Examples are given to show that R [[x]] need not be c-commutative even if R[x] is, Finally, examples are given to answer Taft's question for the special case of a zero-commutative ring.

scholarly journals Ideals in simple rings

1978 ◽  
Vol 25 (3) ◽  
pp. 322-327
Author(s):  
W. Harold Davenport
Keyword(s):  
Lie Algebra ◽  
Associative Ring ◽  
Alternative Ring ◽  
Cayley Algebra ◽  
Characteristic 3 ◽  
Associative Rings ◽  
Algebra Over A Field

AbstractIn this article, we define the concept of a Malcev ideal in an alternative ring in a manner analogous to Lie ideals in associative rings. By using a result of Kleinfield's we show that a nonassociative alternative ring of characteristic not 2 or 3 is a ring sum of Malcev ideals Z and [R, R] where Z is the center of R and [R, R] is a simple non-Lie Malcev ideal of R. If R is a Cayley algebra over a field F of characteristic 3 then [R, R] is a simple 7 dimensional Lie algebra. A similar result is obtained if R is a simple associative ring.

FINITE HIGHER COMMUTATORS IN ASSOCIATIVE RINGS

2013 ◽  
Vol 89 (3) ◽  
pp. 503-509
Author(s):  
CHARLES LANSKI
Keyword(s):  
Nilpotent Element ◽  
Associative Ring ◽  
Minimal Cardinality ◽  
Associative Rings ◽  
The Ideal

AbstractIf $T$ is any finite higher commutator in an associative ring $R$, for example, $T= [[R, R] , [R, R] ] $, and if $T$ has minimal cardinality so that the ideal generated by $T$ is infinite, then $T$ is in the centre of $R$ and ${T}^{2} = 0$. Also, if $T$ is any finite, higher commutator containing no nonzero nilpotent element then $T$ generates a finite ideal.

scholarly journals Some functors arising from the consideration of torsion theories over noncommutative rings

1976 ◽  
Vol 15 (3) ◽  
pp. 455-460 ◽  
Author(s):  
Jonathan S. Golan
Keyword(s):  
Distributive Lattice ◽  
Commutative Ring ◽  
Commutative Rings ◽  
Noncommutative Rings ◽  
Torsion Theories ◽  
Associative Rings

To each associative (but not necessarily commutative) ring R we assign the complete distributive lattice R-tors of (hereditary) torsion theories over R-mod. We consider two ways of making this process functorial – once contravariantly and once covariantly – by selecting appropriate subcategories of the category of associative rings. Combined with a functor due to Rota, this gives us functors from these subcategories to the category of commutative rings.

scholarly journals Direct Product of Finite Intuitionistic Anti Fuzzy Normal Subrings over Non-associative Rings

2019 ◽  
Vol 12 (2) ◽  
pp. 622-648 ◽  
Author(s):  
Nasreen Kausar
Keyword(s):  
Direct Product ◽  
Associative Ring ◽  
Intuitionistic Fuzzy ◽  
Associative Rings

Shal et. al cite:SKR, have introduced the concept of intuitionistic fuzzy normal subrings over a non-associative ring. In this paper, we investigate the concept of intuitionistic anti fuzzy normal subrings over non-associative rings and give some properties of such subrings

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