We show that the category of discrete modules over an infinite profinite group has no non-zero projective objects and does not satisfy Ab4*. also prove same types results in a generalized setting using ring with linear topology.

An element a in ring R has gs-Drazin inverse if there exists b ? comm2(a) such that = b2a, ab Rqnil. For any s C(R), we completely determine when generalized matrix A Ks(R) over local inverse.

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 ...

Odd degree Chebyshev polynomials over a ring of modulo 2 have two kinds of period. One is an “orbital period”. Odd degree Chebyshev polynomials are bijection over the ring. Therefore, when an odd degree Chebyshev polynomial iterate affecting a factor of the ring, we can observe an orbit over the ring. The “ orbital period ” is a period of the orbit. The other is a “degree period”. It is observe...

given a mapping $f:xto x$ we naturally associate to it a monotonic map $gg_f:exp xto exp x$ from the power set of $x$ into itself, thus inducing a generalized topology on $x$. in this paper we investigate some properties of generalized topologies which are defined by such a procedure.

The Schur polynomials s ? are essential in understanding the representation theory of general linear group. They also describe cohomology ring Grassmannians. For ?=(n,n-1,?,1) a staircase shape and ??? subpartition, Stembridge equality states that ?/? =s T . This provides information about symmetry ring. stable Grothendieck G , dual g developed by Buch, Lam, Pylyavskyy, variants K-theory Using ...

In this paper we improve the bounds on the complexity of solutions to the ̄ring squad problem, also known as the ̄ring synchronization problem. In the ̄ring synchronization problem we consider a one-dimensional array of n identical ̄nite automata. Initially all automata are in the same state except for one automaton designated as the initiator for the synchronization. Our results hold for the origi...

Let R be a Cohen-Macaulay local ring of dimension d with infinite residue field. Let I be an R-ideal that has analytic spread `(I) = d, Gd condition and the Artin-Nagata property AN − d−2. We provide a formula relating the length λ(In+1/JIn) to the difference PI(n)−HI(n), where J is a general minimal reduction of I, PI(n) and HI(n) are the generalized Hilbert-Samuel polynomial and the generaliz...

Let R be a ring with involution ′∗′. A map δ of the ring R into itself is called a derivation if δ(xy) = δ(x)y + xδ(y) for all x, y ∈ R. An additive map F : R → R is called a generalized derivation on R if F(xy) = F(x)y + xδ(y) for all x, y ∈ R, Permanent address: Department of Mathematics, Faculty of Science, Aligarh Muslim University, Aligarh202002, India 292 Shakir Ali and Husain Alhazmi whe...

In this paper, we study skew constacyclic codes over the ring ZqR where R = Zq + uZq, q = p s for a prime p and u2 = 0. We give the definition of these codes as subsets of the ring ZqR . Some structural properties of the skew polynomial ring R[x, θ] are discussed, where θ is an automorphism of R. We describe the generator polynomials of skew constacyclic codes over R and ZqR. Using Gray images ...