T. Dupuy, E. Katz, J. Rabinoff, D. Zureick-Brown introduced the module of total $p$-differentials for a ring over $Z/p^2Z$. We study same construction $Z_{(p)}$ and prove regularity criterion. For local ring, tensor product with residue field is constructed in different way by O. Gabber, L. Ramero. In another article arXiv:2006.00448, we use sheaf FW-differentials to define cotangent bundle mic...

in this work, we investigate the transfer of some homological properties from a ring $r$ to its amalgamated duplication along some ideal $i$ of $r$ $rbowtie i$, and then generate new and original families of rings with these properties.

It is proved that a ring is periodic if and only if, for any elements x and y , there exist positive integers k,l,m, and n with either k =m or l =n, depending on x and y , for which xkyl = xmyn. Necessary and sufficient conditions are established for a ring to be a direct sum of a nil ring and a J-ring. 2000 Mathematics Subject Classification. Primary 16U99, 16N20, 16D70.

for a ring endomorphism $alpha$ and an $alpha$-derivation $delta$, we introduce a concept, so called skew $pi$-armendariz ring, that is a generalization of both $pi$-armendariz rings, and $(alpha,delta)$-compatible skew armendariz rings. we first observe the basic properties of skew $pi$-armendariz rings, and extend the class of skew $pi$-armendariz rings through various ring extensions. we nex...

Let A be a ring of dimension d containing an infinite field k, T1,…,Tr variables over and P projective A[T1,…,Tr]-module rank n. Assume one the following conditions holds. 2n≥d+3 is extended from A. 2n≥d+2, affine F‾p-algebra singular locus Spec(A) closed set V(J) with ht J≥d−n+2.

We consider both gauged and ungauged minimal supergravities in five dimensions analyse the charged rotating solutions with two equal angular momenta J. When electric charge Q ∼ J2/3 some specific coefficient, we find new extremal black objects emerge that are asymptotic to either Minkowski or global AdS spacetimes can be best described as degenerate rings. Their near-horizon geometry is locally...

For an element $r$ of a ring $R$, Diophantine $D(r)$ $m$-tuple is $(a_1,a_2,\ldots,a_m)$ elements $R$ such that for all $i,j$ with $i\neq j$, $a_ia_j+r$ perfect square in $R$. In this article, we compute and estimate the measures sets $m$-tuples $\mathbb{Z}_p$ $p$-adic integers, as well its residue field $\mathbb{F}_p$.

in this paper, we introduce the new notion of n-f-clean rings as a generalization of f-clean rings. next, we investigate some properties of such rings. we prove that mn(r) is n-f-clean for any n-f-clean ring r. we also, get a condition under which the denitions of n-cleanness and n-f-cleanness are equivalent.

a topoframe, denoted by $l_{ tau}$,  is a pair $(l, tau)$ consisting of a frame $l$ and a subframe $ tau $ all of whose elements are complementary elements in$l$. in this paper, we define and study the notions of a$tau $-real-continuous function on a frame $l$ and the set of realcontinuous functions $mathcal{r}l_tau $ as an $f$-ring.we show that $mathcal{r}l_{ tau}$is actually a generalization ...