we prove that each normal automorphism of the $n$-periodic product of cyclic groups of odd order $rge1003$ is inner, whenever $r$ divides $n$.

we prove that each normal automorphism of the $n$-periodic product of cyclic groups of odd order $rge1003$ is inner, whenever $r$ divides $n$.

Let R be an integral domain. Note some obvious facts. Every ele­ ment a of R divides 0. Indeed 0 = 0 · a. On the other hand, 0 only divides 0. Indeed if a = q · 0, then a = 0 (obvious!). Finally every unit u divides any other element a. Indeed if v ∈ R is the inverse of u, so that uv = 1 then a = a · 1 = (av)u. Lemma 19.3. Let R be an integral domain and let p ∈ R. Then p is prime if and only i...

Let $R$ be a commutative ring and $M$ be an $R$-module. In this paper, we investigate some properties of 2-absorbing submodules of $M$. It is shown that $N$ is a 2-absorbing submodule of $M$ if and only if whenever $IJLsubseteq N$ for some ideals $I,J$ of R and a submodule $L$ of $M$, then $ILsubseteq N$ or $JLsubseteq N$ or $IJsubseteq N:_RM$. Also, if $N$ is a 2-absorbing submodule of ...

The first part of the paper is concerned to relationship between the sets of associated primes of the generalized $d$-local cohomology modules and the ordinary  generalized local cohomology  modules.  Assume that $R$ is a commutative Noetherian local ring, $M$ and $N$  are  finitely generated  $R$-modules and $d, t$ are two integers. We prove that $Ass H^t_d(M,N)=bigcup_{Iin Phi} Ass H^t_I(M,N)...

Let In = {1; 2; : : : ; n} and x : In → R be a map such that ∑i∈In xi ¿ 0. (For any i, its image is denoted by xi.) Let F = {J ⊂ In : |J | = k, and ∑j∈J xj ¿ 0}. Manickam and Singhi (J. Combin. Theory Ser. A 48 (1) (1988) 91–103) have conjectured that |F|¿ ( n−1 k−1 ) whenever n¿ 4k and showed that the conclusion of the conjecture holds when k divides n. For any two integers r and ‘ let [r]‘ de...

We show that if 4 divides I, then any partial triple system of order r and index 1 can be embedded in a proper triple system of index I and order n whenever n is I-admissible and n > 2r + 1. Moreover we find a set of necessary conditions for the embedding of a partial triple system of index I when I is even and show that when 4 divides 1, then a very closely related set of conditions is suffici...

‎‎Let $R$ be a multiplicative hyperring‎. In this paper‎, ‎we introduce and study the concept of n-absorbing hyperideal which is a generalization‎ ‎of prime hyperideal‎. ‎A proper hyperideal $I$ of $R$ is called an $n$-absorbing hyperideal of ‎$‎R‎$‎ if whenever $alpha_1o...oalpha_{n+1} subseteq I$ for $alpha_1,...,alpha_{n+1} in R$‎, ‎then there are $n$ of the $alpha_i^,$s whose product ...

Let R be an associative ring with identity, C(R) be the category of com-plexes of R-modules and Flat(C(R)) be the class of all at complexes of R-modules. We show that the at cotorsion theory (Flat(C(R)); Flat(C(R))−)have enough injectives in C(R). As an application, we prove that for each atcomplex F and each complex Y of R-modules, Exti (F,X)= 0, whenever Ris n-perfect and i > n.

let r be an associative ring with identity, c(r) be the category of com-plexes of r-modules and flat(c(r)) be the class of all at complexes of r-modules. we show that the at cotorsion theory (flat(c(r)); flat(c(r))−)have enough injectives in c(r). as an application, we prove that for each atcomplex f and each complex y of r-modules, exti (f,x)= 0, whenever ris n-perfect and i > n.