A note on finite local rings

A Note on Rings of Finite Rank

The rank rk(R) of a ring R is the supremum of minimal cardinalities of generating sets of I as I ranges over ideals of R. Matson showed that every n ∈ Z+ occurs as the rank of some ring R. Motivated by the result of Cohen and Gilmer that a ring of finite rank has Krull dimension 0 or 1, we give four different constructions of rings of rank n (for all n ∈ Z+). Two constructions use one-dimension...

Local Rings of Finite Cohen-macaulay Type

Let (R,m) be a local Cohen-Macaulay ring whose m-adic completion R̂ has an isolated singularity. We verify the following conjecture of F.-O. Schreyer: R has finite Cohen-Macaulay type if and only if R̂ has finite Cohen-Macaulay type. We show also that the hypersurface k[[x0, . . . , xd]]/(f) has finite Cohen-Macaulay type if and only if k [[x0, . . . , xd]]/(f) has finite Cohen-Macaulay type, whe...

a note on finite c-tidy groups

let $g$ be a group and $x in g$‎. ‎the cyclicizer of $x$ is defined to be the subset $cyc(x)=lbrace y in g mid langle x‎, ‎yrangle ; {rm is ; cyclic} rbrace$‎. ‎$g$ is said to be a tidy group if $cyc(x)$ is a subgroup for all $x in g$‎. ‎we call $g$ to be a c-tidy group if $cyc(x)$ is a cyclic subgroup for all $x in g setminus k(g)$‎, ‎where $k(g)$ is the intersection of all the cyclicizers in ...

A Note on א0-injective Rings

A ring R is called right א0-injective if every right homomorphism from a countably generated right ideal of R to RR can be extended to a homomorphism from RR to RR. In this note, some characterizations of א0-injective rings are given. It is proved that if R is semiperfect, then R is right א0injective if and only if every homomorphism from a countably generated small right ideal of R to RR can b...

A Note on /-regularity in Rings