nLab Banach ring

Idea

A Banach ring is a complete normed ring, hence a commutative monoid in the monoidal category of complete normed groups (with short group homomorphisms and the projective tensor product).

If not just in complete normed groups but in complete normed vector spaces (Banach spaces), then this is a Banach algebra.

The Berkovich spectrum of a Banach ring $R$ is the topological space of multiplicative seminorms on $R$ that are bounded by the norm on $R$.

Examples

• The integers $\mathbb{Z}$ equipped with their absolute value norm ${\vert- \vert_\infty}$ are a Banach ring.

• The integers with the $p$-adic norm ${|-|_p}$ are an incomplete normed ring whose completion is the Banach ring $\mathbb{Z}_p$ of $p$-adic integers.

Related concepts

algebraic structure	group	ring	field	vector space	algebra
(submultiplicative) norm	normed group	normed ring	normed field	normed vector space	normed algebra
multiplicative norm (absolute value/valuation)			valued field
completeness	complete normed group	Banach ring	complete field	Banach vector space	Banach algebra

References

A quick review is in

• Sarah Brodsky, Non-archimedean geometry (pdf)

A standard textbook account in the context of rigid analytic geometry is

• Siegfried Bosch, Ulrich Güntzer, Reinhold Remmert, section 1.2.4 of Non-Archimedean Analysis – A systematic approach to rigid analytic geometry, 1984 (pdf)

A set of lecture notes in the context of Berkovich spaces is

• Vladimir Berkovich, section 1.2 of Non-archimedean analytic spaces, lectures at the Advanced School on $p$-adic Analysis and Applications, ICTP, Trieste, 31 August - 11 September 2009 (pdf)

Discussion from a more topos-theoretic point of view is in

• Oren Ben-Bassat, Kobi Kremnizer, section 6.5 of Non-Archimedean analytic geometry as relative algebraic geometry (arXiv:1312.0338)

category: analysis