PRESIMPLIFIABLE AND WEAKLY PRESIMPLIFIABLE RINGS

Abstract

Let be a commutative ring with identity. Two elements and b in are called to be associates if and, or equivalently, if. The generalization of associate relation in R has given the idea for definitions of presimplifiable and weakly presimplifiable rings. First of all, it will be given definitions of very strong associate relation, strong regular associate relation, very strongly associate ring, and strongly regular associate ring. The presimplifiable ring is a commutative ring with the condition that every nonzero element is a unit element. While the weakly presimplifiable ring is a commutative ring with the condition that every nonzero element is regular element. Furthermore, it is shown that the relationship between very strongly associate ring with presimplifiable ring and the linkage between strongly regular associate ring and weakly presimplifiable ring. It is obtained that is a presimplifiable ring if and only if is a very strongly associate ring. Meanwhile, is a weakly presimplifiable ring if and only if is a strongly regular associate ring. Then, it is shown that the correlation between presimplifiable and weakly presimplifiable rings to its polynomial ring and its the formal power series ring. If is a weakly presimplifiable ring, then and are also weakly presimplifiable rings. However, if is a presimplifiable ring, then is also a presimplifiable ring but always not valid for.