Split-null extensions of strongly right bounded rings

A ring is said to be strongly right bounded if every nonzero right ideal contains a nonzero ideal. In this paper strongly right bounded rings are characterized, conditions are determined which ensure that the split-null (or trivial) extension of a ring is strongly right bounded, and we characterize strongly right bounded right quasi-continuous split-null extensions of a left faithful ideal over a semiprime ring . This last result partially generalizes a result of C. Faith concerning split-null extensions of commutative FPF rings

Ring hulls and applications

AbstractOur goal is to develop methods that enable one to select a class K of rings and then to describe all right essential overrings or all right rings of quotients of a given ring R which lie in K. Our major method of attack is to determine the existence and/or uniqueness of right ring hulls of R in K and to use these to characterize the right essential overrings of R which are in K. Some applications are: (1) a characterization of the right rings of quotients of the 2-by-2 upper triangular matrix ring over a PID which are either Baer or right extending; (2) a characterization of a continuous ring hull for a commutative ring whose singular ideal has finite uniform dimension; (3) a characterization of the right extending rings which have the 2-by-2 matrix ring over a given division ring for their maximal right ring of quotients; (4) a characterization of the intermediate right extending rings between the 2-by-2 upper triangular matrix ring and the 2-by-2 matrix ring over a large class of local right finitely Σ-extending rings; (5) a characterization of the classical right ring of quotients as a ring hull from a certain class of rings

Quasi-projective modules and the finite exchange property

We define a module M to be directly refinable if whenever M=A+B, there exists A¯⊆A and B¯⊆B such that M=A¯⊕B¯ . Theorem. Let M be a quasi-projective module. Then M is directly refinable if and only if M has the finite exchange property

Split-null extensions of strongly right bounded rings

A ring is said to be strongly right bounded if every nonzero right ideal contains a nonzero ideal. In this paper strongly right bounded rings are characterized, conditions are determined which ensure that the split-null (or trivial) extension of a ring is strongly right bounded, and we characterize strongly right bounded right quasi-continuous split-null extensions of a left faithful ideal over a semiprime ring . This last result partially generalizes a result of C. Faith concerning split-null extensions of commutative FPF rings