Abstract We provide a proof of Jacobson’s theorem on derivations primitive rings with nonzero socle. Both and its formulation (in terms the so-called differential operators left vector spaces over division ring) underlie our paper. apply to describe standard operator real, complex, or quaternionic normed spaces. Indeed, when space is infinite-dimensional, every derivation such ring form $$A\rig...

let $r$ be a reversible ring which is $alpha$-compatible for an endomorphism $alpha$ of $r$ and $f(x)=a_0+a_1x+cdots+a_nx^n$ be a nonzero skew polynomial in $r[x;alpha]$. it is proved that if there exists a nonzero skew polynomial $g(x)=b_0+b_1x+cdots+b_mx^m$ in $r[x;alpha]$ such that $g(x)f(x)=c$ is a constant in $r$, then $b_0a_0=c$ and there exist nonzero elements $a$ and $r$ in $r$ such tha...

we observe some new characterizations of $n$-presented modules. using the concepts of $(n,0)$-injectivity and $(n,0)$-flatness of modules, we also present some characterizations of right $n$-coherent rings, right $n$-hereditary rings, and right $n$-regular rings.

in this paper, we introduce the concept of the   generalized aip-rings as a generalization of the generalized quasi-   baer rings   and generalized p.p.-rings. we show that the class of the generalized aip-rings is closed under direct products and morita invariance. we also characterize the 2-by-2 formal upper triangular matrix rings of this new class of rings. finally, we provide sever...

this paper is an attempt to generalize, simultaneously, the ring of real-valued continuous functions and the ring of real-valued measurable functions.

where the a,are r fixed nonzero elements of Z, and 0<«i(a) <Ui(a) (i = 2, ■ ■ • , r), then A=Z. In [3, Theorem 11; 5; 2; l] specialized forms of (1) (e.g. cn(o) £Z, an(-a) —aEZ) are shown to imply commutativity at least for semi-simple rings. It is natural therefore to seek an extension of Nakayama's result to semi-simple rings. Since a semi-simple ring is a subdirect sum of primitive rings and...

SONET/WDM networks using wavelength add–drop multiplexing can be constructed using certain graph decompositions used to form a “grooming,” consisting of unions of certain primitive rings. The existence of such decompositions when every pair of sites employs no more than 8 of the wavelength capacity is determined, with few possible exceptions, when the ring size is a multiple of four. The techni...

SONET/WDM networks using wavelength add–drop multiplexing can be constructed using certain graph decompositions used to form a “grooming,” consisting of unions of certain primitive rings. The existence of such decompositions when every pair of sites employs no more than 8 of the wavelength capacity is determined, with few possible exceptions, when the ring size is a multiple of four. The techni...

let $r$ be an associative ring with identity and $z^*(r)$ be its set of non-zero zero divisors.  the zero-divisor graph of $r$, denoted by $gamma(r)$, is the graph whose vertices are the non-zero  zero-divisors of  $r$, and two distinct vertices $r$ and $s$ are adjacent if and only if $rs=0$ or $sr=0$.  in this paper, we bring some results about undirected zero-divisor graph of a monoid ring ov...

Isolation is a long-standing security challenge. Privilege rings and virtual memory are increasingly augmented with capabilities, protection keys, powerful enclaves. Moreover, we facing an increased need for physical protection, e.g., via transparent encryption, resulting in complex interplay of various mechanisms. In this work, tackle the isolation challenge new extensible primitive called aut...