We define non-pure commutative rings as a generalization of commutative ringsand rings which admit autodual endomorphisms. A main result in this paper is that a commutative ring whose order is finite and injective as a module over itself is pure.
In this paper we give necessary and sufficient conditions for a fuzzy subring or a fuzzy integral subring A of a commutative ring R to be extendable to one Ae of a commutative ring S containing R as a subring such that A and Ae have the same image. One of the applications of these results gives a criterion for a fuzzy subring of an integral domain R to be extendable to a fuzzy subfield of the quotient field of R.
The present paper studies some properties of local noetherian valuation domains specially the theorbo local noetherian valuation domain which forms the underlying ring of algebras involving torsion. In the concluding chapter of this dissertation, I hope to provide an example of how to construct a resolution of a torsion group structure on a local regular valuation domain in a way that is, hopefully, as painless as possible. 7211a4ac4a