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For which (commutative, unital) rings $R$ is it the case that $R^*\cup \{0\}$ is a subring of $R$? Here are the only examples I could think of:

  • a field
  • $R[x_1,\ldots,x_n]$ for $R$ with the above property
  • $(\mathbb F_2)^\alpha$ for any $\alpha$; here the only unit is $1=(1,1,\ldots)$. This doesn't work for any other field, because if $a\in k^*$, $a\neq 1$, then $(a,1,1,\ldots)-(1,1,1,\ldots)$ is not a unit.
  • Any other ring with characteristic 2 and no units besides 1 (is there any way to categorize these?)

Are there any others?

Alex Mine
  • 188

1 Answers1

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In your notation $k=R^*\cup\{0\}$ is a subfield of the ring $R$. So, you are asking for $k$-algebras $R$ with the property that $R^*=k^*$. There are many rings with this property.

For any smooth projective absolutely irreducible algebraic curve $C$ over $k$ and any point $P$ of $C$, consider the subring $R$ of the function field $C(k)$ that consists of functions without poles outside $P$. The ring $R$ has the property that $R^*=k^*$.

Rene Schoof
  • 839