Endomorphism rings of modules appear as the center of a ring, as the fix ring of a ring with group action or as the subring of constants of a derivation. This note discusses the question whether certain ∗-prime modules have a prime endomorphism ring. Several conditions are presented that guarantee the primeness of the endomorphism ring. The contours of a possible example of a ∗-prime module who...

For p an odd prime, the cohomology ring of the extraspecial p-group of order p5 and exponent p is investigated. A presentation is obtained for the subquotient generated by Chern classes, modulo nilradical. Moreover, it is proved that, for every extraspecial p-group of exponent p, the top Chern classes of the irreducible representations do not generate the Chern subring modulo nilradical. Finall...

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 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...