It is proved that whenever P is a prime ideal in a commutative Noethe-rian ring such that the P-adic and the P-symbolic topologies are equivalent, then the two topologies are equivalent linearly. Several explicit examples are calculated, in particular for all prime ideals corresponding to non-torsion points on nonsingular elliptic cubic curves. There are many examples of prime ideals P in commu...

Let φ : (R, m)→ (S, n) be a local homomorphism of commutative noetherian local rings. Suppose that M is a finitely generated S-module. A generalization of Grothendieck’s non-vanishing theorem is proved for M (i.e. the Krull dimension of M over R is the greatest integer i for which the ith local cohomology module of M with respect to m, Hi m(M), is non-zero). It is also proved that the Gorenstei...

The purpose of this paper is to outline a new approach to the classii-cation of nitely generated indecomposable modules over certain kinds of pullback rings. If R is the pullback of two hereditary noetherian serial rings over a common semi{simple artinian ring, then this classiication can be divided into the classiica-tion of indecomposable artinian modules and those modules over the coordinate...

Definition 2.1. A commutative ring R is Noetherian if every chain of ideals in R I0 ⊂ I1 ⊂ I2 ⊂ · · · terminates after a finite number of steps (i.e., there is an interger k such that Is = Ik if s ≥ k). Remark 2.2. Polynomial rings R [X1, . . . , Xn] are Noetherian if R is. In particular, S (p) = C [p∗] is Noetherian. Theorem 2.3. If R is a Noetherian ring, and M is a finitely generated R-modul...

 ‎In this article‎, ‎we first‎ ‎show that non-Noetherian Artinian uniserial modules over‎ ‎commutative rings‎, ‎duo rings‎, ‎finite $R$-algebras and right‎ ‎Noetherian rings are $1$-atomic exactly like $Bbb Z_{p^{infty}}$‎. ‎Consequently‎, ‎we show that if $R$ is a right duo (or‎, ‎a right‎ ‎Noetherian) ring‎, ‎then the Noetherian dimension of an Artinian‎ ‎module with homogeneous uniserial dim...