Pointwise maximal subrings

Abstract

Let R be a commutative ring with identity. We study the concept of pointwise maximal subrings of a ring. A ring R is called a pointwise maximal subring of a ring T if $$R\subset T$$ and for each $$t\in T{\setminus } R$$ , the ring extension $$R[t]\subseteq T$$ has no proper intermediate ring. A characterization of local, integrally closed pointwise maximal subrings of a ring is given. Let G be a subgroup of the group of automorphisms of T. Then the integrally closed pointwise maximality is a G-invariant property of ring extension under some conditions. We also discuss the number of overrings and the Krull dimension of pointwise maximal subrings of a ring. The pointwise maximal subrings of the polynomial ring R[X] are also discussed.