Pointwise maximal subrings
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Pointwise maximal subrings Rahul Kumar1 · Atul Gaur1 Received: 25 August 2020 / Accepted: 5 October 2020 © The Managing Editors 2020
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 ⊂ T and for each t ∈ T \R, the ring extension R[t] ⊆ 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. Keywords Pointwise maximal subring · Valuation domain · Integrally closed rings · Normal pair · Ring of invariants Mathematics Subject Classification Primary 13B99 · 13A50; Secondary 13B22 · 13A18
1 Introduction All rings considered below are commutative with nonzero identity; all ring extensions, ring homomorphisms, and algebra homomorphisms are unital. By an overring of R, we mean a subring of the total quotient ring of R containing R. By a local ring, we mean a ring with unique maximal ideal. The symbol ⊆ is used for inclusion, while ⊂ is used for proper inclusion. Throughout this paper, qf(R) denotes the quotient field
R. Kumar: The author was supported by the SRF Grant from UGC India, Sr. No. 2061440976. A. Gaur: The author was supported by the MATRICS Grant from DST-SERB, No. MTR/2018/000707.
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Atul Gaur [email protected] Rahul Kumar [email protected]
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Department of Mathematics, University of Delhi, Delhi, India
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Beitr Algebra Geom
of an integral domain R and R denotes the integral closure of R in qf(R). A ring R is said to be a maximal subring of a ring T (equivalently, R ⊂ T is a minimal ring extension, see Ferrand and Olivier 1970) if R ⊂ T and the ring extension R ⊂ T has no proper intermediate ring, see Azarang (2009). A natural question is to think of those ring extensions R ⊂ T , where R may not be a maximal subring of T ; however R[t] is a maximal subring of T for any t ∈ T \R. These ring extensions were studied by Cahen et al. (2018). A ring extension R ⊂ T is said to be a co-pointwise minimal extension if for each t ∈ T \R, R[t] is a maximal subring of T , see Cahen et al. (2018). In Cahen et al. (2018, Lemma 3.8), Cahen at al. introduced a ring extension R ⊂ T where for each t ∈ T \R, R[t] ⊆ T has no intermediate ring. In this case, we call R is a pointwise maximal subring of T . Thus, a ring R is called a pointwise maximal subring of a ring T if R ⊂ T and for each t ∈ T \R, the ring extension R[t] ⊆ T has no proper intermediate ring. Note that if R ⊂ T co-pointwise minimal extension, then R is a pointwise maximal subring of T . However, the converse is not true, for example, take Z ⊂ Z[1/2, 1/3]. In this paper, we establish some characteriza
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