NOTE ON THE DW RINGS

Note on Star Operations over Polynomial Rings

This paper studies the notions of star and semistar operations over a polynomial ring. It aims at characterizing when every upper to zero in R[X] is a ∗-maximal ideal and when a ∗-maximal ideal Q of R[X] is extended from R, that is, Q = (Q ∩ R)[X] with Q ∩R 6= 0, for a given star operation of finite character ∗ on R[X]. We also answer negatively some questions raised by Anderson-Clarke by const...

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...

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...

Note on Subdirect Sums of Rings

where (a) denotes the two-sided ideal of R generated by a. Then J is the Jacobson radical [6 ] of R, and N is the radical of R as defined in [l]. It is well known that J = 0 if and only if R is isomorphic to a subdirect sum of primitive rings, and ^ = 0 if and only if R is isomorphic to a subdirect sum of simple rings with unit element. The above definitions of / and N suggest that it might be ...

A Note on /-regularity in Rings