A ring R is uniquely (nil) clean in case for any $a in R$ there exists a uniquely idempotent $ein R$ such that $a-e$ is invertible (nilpotent). Let $C =(A V W B)$ be the Morita Context ring. We determine conditions under which the rings $A,B$ are uniquely (nil) clean. Moreover we show that the center of a uniquely (nil) clean ring is uniquely (nil) clean.

We construct stationary solutions possessing two vortex rings to the nonlinear Schrödinger type problem iut = −ε4u + ( V + |u| ) u, where the unknown function u is defined as u : R3×R→ C, ε is a small positive parameter and V is a real smooth potential. These two vortex rings will be pinned at a fixed site by the potential V . They lie in the same plane and have neighboring interaction in the n...

We first explain the historical and logical relations of hyperstructures introduced by M. Krasner and R. Rota, and generalized by T. Vougiouklis. Then, with our new algorithm based on our previous results on hypergroups and H v-groups of order 2, 3 and 4, we enumerate hyperrings and H v-rings. More precisely, we found 63 hyperrings of order 2, 875 H v-rings of order 2 and 33,277,642 hyperrings ...

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...

For an o-minimal expansion R of a real closed eld and a set V of Th(R)-convex valuation rings, we construct a \pseudo completion" with respect to V. This is an elementary extension S of R generated by all completions of all the residue elds of the V 2 V , when these completions are embedded into a big elementary extension of R. It is shown that S does not depend on the various embeddings up to ...

We study the properties of F-rationality and F-regularity in multigraded rings and their diagonal subalgebras. The main focus is on diagonal subalgebras of bigraded rings: these constitute an interesting class of rings since they arise naturally as homogeneous coordinate rings of blow-ups of projective varieties. LetX be a projective variety over a fieldK, with homogeneous coordinate ringA. Let...

This paper studies wavelength assignment algorithms on WDM all-optical trees of rings under different models: static, incremental and dynamic. It is shown that 5L/2 is the tight bound of the number of required wavelengths for static trees of rings with load L. This paper also proposes an O[log2(t+1)]-approximation and a  ∑ = ∈ h i R r r V i 1 | ) ( | log max   +h-approximation algorithm for ...

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