If R is a local ring whose radical J(R) is nilpotent and R/J(R) is a commutative field which is algebraic over GF(p), then R has at least one subring S such that S w*=,S,, where each S, is isomorphic to a Galois ring and S/J(S) is naturally isomorphic to R/J(R). Such subrings ofR are mutually isomorphic, but not necessarily conjugate in R.

Motivated by symbolic dynamics, we study the problem, given a unital subring 5 of the reals, when is a matrix A algebraically shift equivalent over S to a primitive matrix? We conjecture that simple necessary conditions on the nonzero spectrum of A are sufficient, and establish the conjecture in many cases. If S is the integers, we give some lower bounds on sizes of realizing primitive matrices...

Let K be a one-variable function field over a field of constants of characteristic 0. Let R be a holomorphy subring of K, not equal to K. We prove the following undecidability results for R: If K is recursive, then Hilbert’s Tenth Problem is undecidable in R. In general, there exist x1, . . . , xn ∈ R such that there is no algorithm to tell whether a polynomial equation with coefficients in Q(x...

For a finite-dimensional Hopf algebra H over a field k and an Hcomodule algebra A, we study properties of A which are preserved when A is twisted by a Hopf 2-cocycle σ on H. We prove that if there exists σ such that Aσ is super-commutative, then A being affine imples that A is Noetherian. If also Hσ is commutative, then A is integral over a central subring of AcoH . We also consider when A sati...

‎Let $f:Arightarrow B$ and $g:A rightarrow C$ be two ring homomorphisms and let $J$ and $J^{'}$ be two ideals of $B$ and $C$‎, ‎respectively‎, ‎such that $f^{-1}(J)=g^{-1}(J^{'})$‎. ‎The bi-amalgamation of $A$ with $(B,C)$ along $(J,J^{'})$ with respect of $(f,g)$ is the subring of $Btimes C$ given by‎ ‎$Abowtie^{f,g}(J,J^{'})={(f(a)+j,g(a)+j^{'})‎/ ‎a in A‎, ‎(j,j^{'}) in Jtimes J^{'}}.$‎ ‎In ...

Let R be a subring of the rationals. We want to investigate self splitting R-modules G that is ExtR(G,G) = 0 holds. For simplicity we will call such modules splitters, see [16]. Also other names like stones are used, see a dictionary in Ringel’s paper [14]. Our investigation continues [10]. In [10] we answered an open problem by constructing a large class of splitters. Classical splitters are f...

this article introduces a new tri-parameter law of corresponding states from the known regularities, namely the lir and zeno line for dense fluids (when tz=tc). two of these parameters are the zeno line parameters and the third parameter (the fluid parameter), has been obtained from the lir, this parameter is solely dependent upon the fluid characteristics. the third parameter can be obtaind fr...

This section define a level subring or level ideals obtain a set of necessary and sufficient condition for the equality of two ideals and characterizes field in terms of its fuzzy ideals. It also presents a procedure to construct a fuzzy subrings (fuzzy ideals) from any given ascending chain of subring ideal. We prove that the lattice of fuzzy congruence of group G (respectively ring R) is isom...

Earlier we justified our proof of the fundamental theorem of arithmetic by arguing that the result was not obvious, since its analogue in the ring E of even integers did not hold. Here we will explore a far more exciting explanation: essentially the same proof can be used to show that certain other “integer-like rings” have the unique factorization property. The unique factorization property ca...