Let k > 1 be an integer and let p be a prime. We show that if p a k < 2p a or k = p a q + 1 (with 2q p) for some a = 1, 2, 3,. .. , then the set { ` n k ´ : n = 0, 1, 2,. .. } is dense in the ring Z p of p-adic integers, i.e., it contains a complete system of residues modulo any power of p.

Let k be a positive integer and p be a prime. We show that if p a k < 2p a or k = p a q + 1 (with 2q p) for some a = 0, 1, 2,. .. , then the set { ` n k ´ : n = 0, 1, 2,. .. } is dense in the ring Z p of p-adic integers, i.e., it contains a complete system of residues modulo any power of p.

we consider the class $mathfrak m$ of $bf r$--modules where $bf r$ is an associative ring. let $a$ be a module over a group ring $bf r$$g$, $g$ be a group and let $mathfrak l(g)$ be the set of all proper subgroups of $g$. we suppose that if $h in mathfrak l(g)$ then $a/c_{a}(h)$ belongs to $mathfrak m$. we investigate an $bf r$$g$--module $a$ such that $g not = g&apos;$, $c_{g}(a) = 1$. we stud...

Let $R$ be a 2-torsion free semiprime ring with extended centroid $C$, $U$ the Utumi quotient ring of $R$ and $m,n>0$ are fixed integers. We show that if $R$ admits derivation $d$ such that $b[[d(x), x]_n,[y,d(y)]_m]=0$ for all $x,yin R$ where $0neq bin R$, then there exists a central idempotent element $e$ of $U$ such that $eU$ is commutative ring and $d$ induce a zero derivation on $(1-e)U$. ...

We recall that Theorem 1.3 allows us to define the ideal class group of a Dedekind domain, and in particular of a ring of integers, as the group of fractional ideals modulo the subgroup of principal ideals. We will prove that in the case of a ring of integers, the ideal class group is finite. In fact, we will shortly give a stronger statement due to Minkowski. Using similar techniques, we will ...

Let k > 1 be an integer and let p be a prime. We show that if pa k < 2pa or k = paq + 1 (with q < p/2) for some a = 1, 2, 3, . . ., then the set { (n k ) : n = 0, 1, 2, . . .} is dense in the ring Zp of p-adic integers; i.e., it contains a complete system of residues modulo any power of p.

We present new algorithms for computing Smith normal forms of matrices over the integers and over the integers modulo d. For the case of matrices over Z Z d, we present an algorithm that computes the Smith form S of an A 2 Z Z nm d in only O(n ?1 m) operations from Z Z d. Here, is the exponent for matrix multiplication over rings: two n n matrices over a ring R can be multiplied in O(n) operati...

Let $R$ be a commutative Noetherian ring with non-zero identity, $fa$ an ideal of $R$, and $X$ an $R$--module. Here, for fixed integers $s, t$ and a finite $fa$--torsion $R$--module $N$, we first study the membership of $Ext^{s+t}_{R}(N, X)$ and $Ext^{s}_{R}(N, H^{t}_{fa}(X))$ in the Serre subcategories of the category of $R$--modules. Then, we present some conditions which ensure the exi...