Sheaf methods applied to coherent rings

A commutative ring is called coherent if the intersection of any two finitely generated ideals is finitely generated and the annihilator ideal of an arbitrary element of the ring is finitely generated. Pierce's representation of a ring R as the ring of all global sections of an appropriate sheaf of rings, k , is described. Some theorems are deduced relating the coherence of the ring R to certain properties of the sheaf k . The sheaves from the above representation for R⌈X⌉ and R⌈⌈G⁺⌉⌉ , where R is a commutative von Neumann regular ring and G is a linearly ordered abelian group, are calculated. Applications of the above theorems now show that R⌈X⌉ is coherent and yield necessary and sufficient conditions for R⌈⌈G⁺⌉⌉ to be coherent.Science, Faculty ofMathematics, Department ofGraduat