Let $R$ be an arbitrary ring and $T$ be a submodule of an $R$-module $M$. A submodule $N$ of $M$ is called $T$-small in $M$ provided for each submodule $X$ of $M$, $Tsubseteq X+N$ implies that $Tsubseteq X$. We study this mentioned notion which is a generalization of the small submodules and we obtain some related results.

Let $M_R$ be a module with $S=End(M_R)$. We call a submodule $K$ of $M_R$ annihilator-small if $K+T=M$, $T$ a submodule of $M_R$, implies that $ell_S(T)=0$, where $ell_S$ indicates the left annihilator of $T$ over $S$. The sum $A_R(M)$ of all such submodules of $M_R$ contains the Jacobson radical $Rad(M)$ and the left singular submodule $Z_S(M)$. If $M_R$ is cyclic, then $A_R(M)$ is the unique ...

We study the p-adic behavior of Jacobi sums for Q(ζp) and link this study to the p-Sylow subgroup of the class group of Q(ζp) + and to some properties of the jacobian of the Fermat curve Xp + Y p = 1 over Fl where l is a prime number distinct from p. Let p be a prime number, p ≥ 5. Iwasawa has shown that the p-adic properties of Jacobi sums for Q(ζp) are linked to Vandiver’s Conjecture (see [5]...

The complete generalized cycle G(d, n) is the digraph which has Zn × Zd as the vertex set and every vertex (i, x) is adjacent to the d vertices (i + 1, y) with y ∈ Zd . As a main result, we give a necessary and sufficient condition for the iterated line digraph G(d, n, k) = Lk−1G(d, n), with d a prime number, to be a Cayley digraph in terms of the existence of a group0d of order d and a subgrou...

It is proved that if R is a valuation domain with maximal ideal P and if RL is countably generated for each prime ideal L, then R R is separable if and only RJ is maximal, where J = ∩n∈NP . When R is a valuation domain satisfying one of the following two conditions: (1) R is almost maximal and its quotient field Q is countably generated (2) R is archimedean Franzen proved in [2] that R is separ...