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...

In this paper we construct a natural category ~r of locally and topologically ringed spaces which contains both the category of locally noetherian formal schemes and the category of rigid analytic varieties as full subcategories. This category has applications in algebraic geometry and rigid analytic geometry. The idea of the definition of the category ~r is the following. From a formal point o...

We study the online page replication problem. We present a new randomized algorithm for rings which is 2.37297-competitive, improving the best previous result of 3.16395. We also show that no randomized algorithm is better than 1.75037-competitive on the ring. This is the rst non-trivial lower bound for the problem. We initiate the study of replication in continuous metric spaces (the Nomads' P...