nLab normed ring

Idea

A normed ring is a ring compatibly equipped with a norm on the underlying abelian group.

If this is suitably complete with respect to the norm, then a normed ring is called a Banach ring. A normed ring which is a field is, naturally, called a normed field, and if the norm is multiplicative it is also called a valued field.

The Berkovich spectrum of a normed ring $R$ is the set of multiplicative seminorms on $R$ that are bounded by the norm on $R$.

Definition

Definition

A normed commutative ring is a commutative ring $R$ equipped with a function

$${\vert -\vert} \;\colon\; R \longrightarrow \mathbb{R}_{\geq 0}$$

to the non-negative real numbers such that for all $f,g \in R$

1. ${\vert f \vert} = 0$ precisely if $f = 0$;

2. $\vert -f\vert = \vert f \vert$

3. ${\vert f + g \vert} \leq {\vert f \vert}+ {\vert g \vert}$ (triangle identity)

4. ${\vert f \cdot g\vert} \leq {\vert f \vert\cdot {\vert g \vert}}$.

(e.g Ozaki, Kashiwagi & Tsuboi 1953)

Remark

One might also define a normed ring to be a commutative monoid internal to the monoidal category $NGrp$ of normed groups. If the morphisms in $NGrp$ are taken to be the short group homomorphisms and the projective cross norm is used on the tensor product, then this reproduces the definition above. If (as is often seen) the morphisms are generalized to bounded group homomrophisms, then this generalizes the third clause in def. to

• there is $C \in \mathbb{R}_{\gt 0}$ such that for all $f,g \in R$

  $${\vert f \cdot g\vert} \leq C \cdot {\vert f \vert\cdot {\vert g \vert}}$$

see e.g. (Bassat-Kremnitzer 13, remark 6.32)

Examples

A normed field is of course in particular a normed ring.

Example

For $R$ a normed commutative ring, then for each $n \in \mathbb{N}$ the matrix algebra $Mat_n(R)$ becomes a normed ring with norm

$${\vert A\vert} \coloneqq max_{1 \leq i,j \leq n}({\vert A_{i,j}\vert}) \,.$$

Notice that even if $R$ if the norm on $R$ is multiplicative (is an absolute value) that on $Mat_n(R)$ is not in general. If $R$ is a Banach ring, then so is $Mat_n(R)$.

(e.g. Jarden 11).

Related concepts

• division ring

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

The notion of normed rings originates with:

• Israel Gelfand, Normierte Ringe, Rec. Math. [Mat. Sbornik] N.S. 9(51) 1 (1941) 3–24 [mathnet:6046]

Early further discussion:

• Shigeo Ozaki, Sadao Kashiwagi, Teruo Tsuboi, Note on Normed Rings, Science Reports of the Tokyo Bunrika Daigaku, Section A 4 98/103 (1953) 277-282 [jstor:43700402]

See also:

• Siegfried Bosch, Ulrich Güntzer, Reinhold Remmert, Non-Archimedean Analysis – A systematic approach to rigid analytic geometry, Grundlehren der Mathem. Wissen. 261 (1984) [ ISBN:9783540125464, pdf]

• Naoki Kimura, A note on normed ring, Kodai Math. Sem. Rep. 1 3-4 (1949) 23-24 [euclid:kmj/1138833472]

• Mark Naimark, Normed Rings, Groningen, Netherlands: P. Noordhoff N. V. (1959)

• M. Jarden, Normed rings, chapter 2 of: Algebraic patching, Springer Monographs in Mathematics (2011) [pdf]

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

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

• ProofWiki, Norm (Division ring)

For more see the references at Banach ring.