## Kyoto Journal of Mathematics

- Kyoto J. Math.
- Volume 59, Number 3 (2019), 703-716.

### Small embeddings of integral domains

#### Abstract

Let $A$ be a geometrically integral algebra over a field $k$. We prove that, for any affine $k$-domain $R$, if there exists an extension field $K$ of $k$ such that $R\subseteq K{\otimes}_{k}A$ and $R\u2288K$, then there exists an extension field $L$ of $k$ such that $R\subseteq L{\otimes}_{k}A$ and ${trdeg}_{k}\left(L\right)<{trdeg}_{k}\left(R\right)$. This generalizes a result of Freudenburg, namely, the fact that this is true for $A={k}^{\left[1\right]}$.

#### Article information

**Source**

Kyoto J. Math., Volume 59, Number 3 (2019), 703-716.

**Dates**

Received: 4 August 2016

Revised: 5 April 2017

Accepted: 16 May 2017

First available in Project Euclid: 21 May 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.kjm/1558404168

**Digital Object Identifier**

doi:10.1215/21562261-2019-0022

**Mathematical Reviews number (MathSciNet)**

MR3990183

**Zentralblatt MATH identifier**

07108008

**Subjects**

Primary: 14R10: Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem)

Secondary: 13B25: Polynomials over commutative rings [See also 11C08, 11T06, 13F20, 13M10] 13G05: Integral domains

**Keywords**

integral domains tensor products ring extensions

#### Citation

Bao, Yu Yang; Daigle, Daniel. Small embeddings of integral domains. Kyoto J. Math. 59 (2019), no. 3, 703--716. doi:10.1215/21562261-2019-0022. https://projecteuclid.org/euclid.kjm/1558404168