Let X be a given set, and let R, S be some classes of certain subsets of X. We say that R is a ring if A+B ∈ R, A−B ∈ R whenever A ∈ R, B ∈ R. We say that S is a σ-ring if it is a ring and if for every sequence A1, A2, . . ., where Ai ∈ S (i = 1, 2, . . .), the sum ∑∞ i=1 Ai is also contained in S. Let B be a commutative Banach algebra with a unity, i.e. a Banach space, where for all pairs f ∈ ...

F(o)F(1)-ATP synthase mediates coupling of proton flow in F(o) and ATP synthesis/hydrolysis in F(1) through rotation of central rotor subunits. A ring structure of F(o)c subunits is widely believed to be a part of the rotor. Using an attached actin filament as a probe, we have observed the rotation of the F(o)c subunit ring in detergent-solubilized F(o)F(1)-ATP synthase purified from Escherichi...

In his 1974 text, Commutative Ring Theory, Kaplansky states that among the examples of non-Dedekind Prüfer domains, the main ones are valuation domains, the ring of entire functions and the integral closure of a Prüfer domain in an algebraic extension of its quotient field [Kap74, p.72]. A similar list today would likely include Kronecker function rings, the ring of integervalued polynomials an...

Let R be a ring with center Z(R). We write the commutator [x, y] = xy− yx, (x, y ∈ R). The following commutator identities hold: [xy,z] = x[y,z] + [x,z]y; [x, yz] = y[x,z] + [x, y]z for all x, y,z ∈ R. We recall that R is prime if aRb = (0) implies that a= 0 or b = 0; it is semiprime if aRa = (0) implies that a = 0. A prime ring is clearly a semiprime ring. A mapping f : R→ R is called centrali...

Let R be a commutative ring and S a multiplicatively closed subset of R having no zero divisors. The pair (.R, S) is said to be stable if the ring of fractions of R, S R, defined by S is a ring for which all finitely generated projective modules are free. For a stable pair (R, S) assume 2 is a unit in R and V is a free R-module of dimension > 3. This paper examines the action of a group automor...

the aim of this paper is to study orders over a valuation ring v with arbitrary rank in acentral simple f-algebra q. the relation between all of the orders is explained with a diagram. it is thenshown that inside bezout order, extremal v-orders are precisely semi-hereditary. in the last section, theeffect of henselization on maximal and semi-hereditary orders is examined.

‎let $mathcal a$ and $mathcal b$ be unital rings‎, ‎and $mathcal m$‎ ‎be an $(mathcal a‎, ‎mathcal b)$-bimodule‎, ‎which is faithful as a‎ ‎left $mathcal a$-module and also as a right $mathcal b$-module‎. ‎let ${mathcal u}=mbox{rm tri}(mathcal a‎, ‎mathcal m‎, ‎mathcal‎ ‎b)$ be the triangular ring and ${mathcal z}({mathcal u})$ its‎ ‎center‎. ‎assume that $f:{mathcal u}rightarrow{mathcal u}$ is...

in this paper, we introduce a class of $j$-quasipolar rings. let $r$ be a ring with identity. an element $a$ of a ring $r$ is called {it weakly $j$-quasipolar} if there exists $p^2 = pin comm^2(a)$ such that $a + p$ or $a-p$ are contained in $j(r)$ and the ring $r$ is called {it weakly $j$-quasipolar} if every element of $r$ is weakly $j$-quasipolar. we give many characterizations and investiga...

Let R be a 2-torsion free prime ring with center Z, right Utumi quotient ring U , generalized derivation F associated with a nonzero derivation d of R and L a Lie ideal of R. If F (uv)n = F (u)mF (v)l or F (uv)n = F (v)lF (u)m for all u, v ∈ L, where m,n, l are fixed positive integers, then L ⊆ Z. We also examine the case where R is 312 Ahmad N. Alkenani et al. a semiprime ring. Finally, as an ...