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normed ring

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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 RR is the set of multiplicative seminorms on RR that are bounded by the norm on RR.

Definition

Definition

A normed commutative ring is a commutative ring RR equipped with a function

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

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

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

  2. |f+g||f|+|g|{\vert f + g \vert} \leq {\vert f \vert}+ {\vert g \vert} (triangle identity)

  3. |fg||f||g|{\vert f \cdot g\vert} \leq {\vert f \vert\cdot {\vert g \vert}}.

e.g (Berkovich 09, def. 1.2.1)

Remark

One might also define a normed ring to be a commutative monoid internal to the monoidal category NGrpNGrp of normed groups. If the morphisms in NGrpNGrp 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 >0C \in \mathbb{R}_{\gt 0} such that for all f,gRf,g \in R

    |fg|C|f||g| {\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 RR a normed commutative ring, then for each nn \in \mathbb{N} the matrix algebra Mat n(R)Mat_n(R) becomes a normed ring with norm

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

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

(e.g. Jarden 11).

algebraic structuregroupringfieldvector spacealgebra
(submultiplicative) normnormed groupnormed ringnormed fieldnormed vector spacenormed algebra
multiplicative norm (absolute value/valuation)valued field
completenesscomplete normed groupBanach ringcomplete fieldBanach vector spaceBanach algebra

References

  • S. Bosch, U. Güntzer, Reinhold Remmert, Non-Archimedean Analysis – A systematic approach to rigid analytic geometry, 1984 (pdf)

  • Naoki Kimura, A note on normed ring, Kodai Math. Sem. Rep. Volume 1, Number 3-4 (1949), 23-24. (Euclid)

  • 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 pp-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.

Last revised on July 18, 2014 at 00:42:14. See the history of this page for a list of all contributions to it.