Minimal convex extensions and intersections of primary I-ideals in f-rings
An f-ring A is said to satisfy the left nth-convexity property if for \(n\in {\mathbb{N}}\) and \(\{u,v\}\subset A_+\) such that \(u\leq v^ n\), there exists a \(w\in A\) such that \(u=wv\). An f-ring satisfies the nth-convexity property if it satisfies both the right and the left nth-convexity properties. Section II studies the embedding of an f-ring in an f-ring with a convexity property. The main results are the following theorems. Theorem 2.2. Let \(n\geq 1\) and A be a semiprime f-ring. Then A has a semiprime n-convexity extension \(A\rightarrowtail B\) iff A can be embedded in a direct product of totally ordered division rings. Theorem 2.4. Let \(n\geq 2\) and A be a commutative semiprime f-ring. Then there is a unique commutative semiprime f-ring \(K_ nA\) which is the smallest n-convexity extension of A and which satisfies the universal mapping property. Section III gives an application showing how the embedding \(A\rightarrowtail K_ nA\) can be used in problems that do not originally mention a convexity property.
scientific article; zbMATH DE number 4106933
scientific article