Roughness in modules have been investigated by B. Davvaz and M. Mahdavipour in 2006 [5]. The purpose of this paper is to introduce and discuss the concept of T -rough prime and primary submodules which is a generalization the lower and upper approximation submodules over a commutative ring. We define a set-valued homomorphism on a module and study some properties of it . We prove some results f...

IN HIS WORK on complex cobordism [ 181 Novikov introduced a spectral sequence converging to the stable homotopy of a space, depending only on the complex cobordism of the space as a module over the ring of primary complex cobordism operations. This spectral sequence can be localized at a prime p and one can work, as Novikov did, with the smaller theory, BP*( ), known as Brown-Peterson cohomolog...

Examples are given to show that the support of a complex of modules over a commutative noetherian ring may not be read off the minimal semi-injective resolution of the complex. These also give examples of semiinjective complexes whose localization need not be homotopically injective. Let R be a commutative noetherian ring. Recall that the support of a finitely generated R-module M is the set of...

We construct certain systems of elements in K2 of some CM elliptic curves. When the classnumber of the field of CM is 1, The image of this system under the regulator map forms an euler system in the sense of Rubin [Ru]. §1. Systems of K2 of CM elliptic curves. Let K be an imaginary quadratic field. Take a Hecke character φ of K of type (1,0) and let fφ be its conductor. Let H = K(fφ) be the ray...

MR0044509 (13,427c) 09.0X Artin, Emil; Tate, John T. A note on finite ring extensions. J. Math. Soc. Japan 3 (1951), 74–77. Let R and S be two commutative rings, R ⊆ S. Then S is called a modulefinite extension of R, if it is an R-module with a finite set of generators, that is, if there exists a finite number of elements of S such that every element of S can be represented as a linear combinat...

a module $m$ is called $emph{h}$-cofinitely supplemented if for every cofinite submodule $e$ (i.e. $m/e$ is finitely generated) of $m$ there exists a direct summand $d$ of $m$ such that $m = e + x$ holds if and only if $m = d + x$, for every submodule $x$ of $m$. in this paper we study factors, direct summands and direct sums of $emph{h}$-cofinitely supplemented modules. let $m$ be an $emph{h}$...

K: a global function field of characteristicp # 0 E a fixed algebraic closure of K 00: a distinguished place of K A: the ring of functions in K regular away from cx> F,: a finite field which is an A-algebra VI the A-characteristic of F, (this means that the prime ideal pt, of v is the kernel of the structure morphism A + F4) 4 : a Drinfeld A-module over F, of rank d 7~: The Frobenius endomorphi...

Let X be a space and write LX for its free loop space equipped with the action of the circle group T given by dilation. We compute the equivariant cohomology H(LXhT;Z/p) as a module over H ∗(BT;Z/p) when X = CP for any positive integer r and any prime number p. The computation implies that the associated mod p Serre spectral sequence collapses from the E3-page. MSC: 55N91; 58E05; 55P35; 18G50

There is a constant C0 such that all nonabelian finite simple groups of rank n over Fq , with the possible exception of the Ree groups G2(3), have presentations with at most C0 generators and relations and total length at most C0(logn + log q). As a corollary, we deduce a conjecture of Holt: there is a constant C such that dimH2(G, M) ≤ C dimM for every finite simple group G, every prime p and ...