Examples of Local Rings | KEVOS® Mathematics

Q: Is a twisted power series ring k[[x;]] ever commutative?

Exactly when is the identity. The relation xa = (a)x shows that x commutes with a iff (a) = a, so nontrivial gives a genuinely noncommutative ring. It is always a local domain with maximal ideal (x) and residue division ring k, so this is the cheapest way to produce a noncommutative local ring.

Q: Why does the exterior algebra fail to be a domain even though it is local?

Because e e = 0 for every e ∈ V, so every degree-one element is a nonzero square-zero element. Locality says nothing about zero divisors: the two conditions are independent. k[[x]] is a local domain, (V) is a local ring with plenty of zero divisors, and Z is a domain that is not local.

Q: How large can the nilpotency index of rad(kG) be for a p-group?

Lam's bound is a^|G| = 0, which is generally not sharp. For G cyclic of order p^n the augmentation ideal is generated by t = g-1 with t^p^n = 0 and t^p^n-1 ≠ 0, so the index is exactly |G| there. For non-cyclic p-groups the index is strictly smaller: for the Klein four group in characteristic 2 the augmentation ideal has cube zero, well below |G| = 4.

Q: Does (19.11) need the group to be finite?

Yes, in two places. Finiteness makes A = RG a finitely generated R-module, which is what lets Nakayama's Lemma apply to a cyclic A-module; and the triviality of the G-action on simple modules in (8.8) is proved for finite p-groups. For infinite locally finite p-groups the augmentation ideal is nil but not nilpotent, and the conclusion has to be argued differently.

Executive Summary

Commutative algebra manufactures local rings by localising at a prime. Noncommutative algebra cannot, so its local rings have to be recognised where they occur. Five families cover almost every example that arises in practice: division rings; power series and twisted power series over a local ring; triangular matrices with constant diagonal; exterior algebras; and group algebras of finite $p$ -groups in characteristic $p$ .

Each is verified the same way — exhibit a two-sided ideal $𝔪$ with $R / 𝔪$ a division ring and $𝔪$ nil or nilpotent, then invoke $(19.1) (3)$ . The differences lie in whether $𝔪$ is nilpotent (completely primary) or merely maximal, and that difference is what separates the artinian examples from the rest.

5Standard families

$𝔪^{n + 1} = 0$ Exterior algebra of

\dim V = n

$| G |$ Nilpotency bound for

k G

$k$ Residue ring of

k G

G

p

-group

Overview

Local Rings: Definition and Equivalent Characterisations reduces locality to a single test: $R \neq 0$ is local exactly when $R / rad R$ is a division ring. This page applies that test repeatedly.

It is worth being explicit about what is not available. Localising a noncommutative ring at a prime ideal requires an Ore condition that generically fails, so there is no functorial machine turning an arbitrary ring into a local one. Every example below is therefore a construction that happens to produce a division-ring quotient, not the output of a localisation.

Find a candidate ideal

𝔪

Check

R / 𝔪

is a division ringCheck

𝔪 \subseteq rad R

𝔪 = rad R

, so

R

is local

The third step is usually the only work, and it is almost always done by showing $𝔪$ is nil or nilpotent and quoting $(4.11)$ : every nil one-sided ideal lies in the radical.

Learning Objectives

Recall the commutative examples — localisations, local rings of varieties, germ rings, valuation rings — and why they do not transfer.
Prove that $R [[x]]$ and $R [[x; σ]]$ are local whenever $R$ is.
Compute $rad A$ for the constant-diagonal triangular matrix ring of $(19.8)$ .
Prove that the exterior algebra of a finite-dimensional vector space is local with nilpotent radical.
State and prove $(19.11)$ for group algebras $R G$ with $G$ a finite $p$ -group.
Separate the completely primary examples from the merely local ones.

Core Concepts

The commutative stock, for orientation

$(19.4)$ Localisation: For a commutative ring $R$ and a prime $𝔭$ , the localisation $R_{𝔭}$ is local with maximal ideal $𝔭 R_{𝔭}$ .
$(19.5)$ Local ring of a variety: At a point $x$ of an algebraic variety $X$ , the rational functions regular at $x$ form a local ring $𝒪_{x}$ ; the maximal ideal is the functions vanishing at $x$ .
$(19.6)$ Germs of holomorphic functions: At a point $x$ of a Riemann surface, germs of functions holomorphic at $x$ form a local ring, with maximal ideal the germs vanishing at $x$ .
$(19.7)$ Valuation rings: Every valuation ring of a field is local. The $p$ -adic integers $ℤ_{p}$ are the rank-one discrete case inside $ℚ_{p}$ , with $rad (ℤ_{p}) = p ℤ_{p}$ .

These four are the intuition. None of them survives verbatim: $(19.4)$ needs Ore localisation, and $(19.5)$ – $(19.6)$ are geometric. What does survive is $(19.7)$ , because the valuation criterion $(19.3) (b)$ makes no use of commutativity.

Power series: the first noncommutative supply

Let $R$ be any ring and $A = R [[x]]$ . A power series is a unit in $A$ exactly when its constant term is a unit in $R$ : if $f = a_{0} - g$ with $a_{0} \in U (R)$ and $g \in x R [[x]]$ , then $f = a_{0} (1 - a_{0}^{- 1} g)$ and $1 - a_{0}^{- 1} g$ is inverted by the geometric series $\sum_{i \geq 0} (a_{0}^{- 1} g)^{i}$ , which converges coefficientwise because $a_{0}^{- 1} g \in x R [[x]]$ .

rad R [[x]] = {a_{0} + a_{1} x + a_{2} x^{2} + \dots : a_{0} \in rad R}

(19.7a)

The variable $x$ always lies in the radical; the constant term is the only obstruction.

Nothing in that argument uses commutativity of $x$ with the coefficients. If $σ \in Aut (R)$ and $A = R [[x; σ]]$ is the twisted power series ring with $x r = σ (r) x$ , the same computation applies verbatim, because $x R [[x; σ]] = R [[x; σ]] x$ is still a two-sided ideal whose powers shrink.

Graded algebras with one-dimensional degree zero

If $A = A_{0} \oplus A_{1} \oplus \dots \oplus A_{n}$ is a graded algebra with $A_{0}$ a division ring and $A_{i} = 0$ for $i > n$ , then $𝔪 = A_{1} \oplus \dots \oplus A_{n}$ is a two-sided ideal with $𝔪^{n + 1} = 0$ and $A / 𝔪 ≅ A_{0}$ . So $A$ is local and completely primary. Exterior algebras $(19.9)$ are the standard instance; so are truncated tensor algebras.

Key Results

Proposition(19.7b)Power series over a local ring

Let $R$ be a ring and $σ$ an automorphism of $R$ . Then $rad R [[x; σ]]$ consists of the series whose constant term lies in $rad R$ , and $R [[x; σ]] / rad R [[x; σ]] ≅ R / rad R$ . In particular, if $R$ is local then so is $R [[x; σ]]$ , with the same residue division ring.

Proof

Write $A = R [[x; σ]]$ and let $𝔘 = {f \in A : f (0) \in rad R}$ , where $f (0)$ denotes the constant term. Since $σ (rad R) = rad R$ (an automorphism preserves the radical) and $f \mapsto f (0)$ is a ring homomorphism $A ↠ R$ , the set $𝔘$ is the preimage of $rad R$ and hence a two-sided ideal, with $A / 𝔘 ≅ R / rad R$ .

For $f \in 𝔘$ , the series $1 - f$ has constant term $1 - f (0) \in 1 + rad R \subseteq U (R)$ , so $1 - f \in U (A)$ by the unit criterion for power series. The same applies to $1 - g f h$ for any $g, h \in A$ , since $𝔘$ is an ideal. By the maximality property of the radical, $𝔘 \subseteq rad A$ .

Conversely, because $𝔘 \subseteq rad A$ , the quotient rule $(4.6)$ gives $rad (A / 𝔘) = (rad A) / 𝔘$ . But $A / 𝔘 ≅ R / rad R$ has zero radical, so $rad A = 𝔘$ . If $R$ is local, $A / rad A ≅ R / rad R$ is a division ring and $(19.1) (3)$ makes $A$ local.

Example(19.8)Constant-diagonal triangular matrices

Let $k$ be a division ring, $T = T_{n} (k)$ the ring of upper triangular $n \times n$ matrices over $k$ , and $J \subseteq T$ the strictly upper triangular matrices. It is standard that $J = rad T$ and $J^{n} = 0$ . Let

A = {a \cdot 1 + N : a \in k, N \in J} = k \cdot 1 \oplus J \subseteq T .

(19.8)

Then $A$ is a subring of $T$ , $J$ is a two-sided ideal of $A$ with $J^{n} = 0$ , and $A / J ≅ k$ . Hence $A$ is local with $rad A = J$ , and completely primary. For $n \geq 3$ it is noncommutative even when $k$ is a field.

Proposition(19.9)Exterior algebras are local

Let $k$ be a field and $V$ a $k$ -vector space with $\dim_{k} V = n < \infty$ . The exterior algebra

R = Λ (V) = k \oplus Λ^{1} (V) \oplus \dots \oplus Λ^{n} (V)

(19.9)

is a local ring with $rad R = 𝔪 = Λ^{1} (V) \oplus \dots \oplus Λ^{n} (V)$ , and $𝔪^{n + 1} = 0$ , $R / 𝔪 ≅ k$ . It is noncommutative precisely when $n \geq 2$ and $char k \neq 2$ .

Proof

$𝔪$ is the ideal of elements with zero component in degree $0$ , so it is two-sided and $R / 𝔪 ≅ k$ is a field. A product of $n + 1$ elements of $𝔪$ lands in degrees $\geq n + 1$ , all of which vanish, so $𝔪^{n + 1} = 0$ . A nilpotent ideal is nil, hence $𝔪 \subseteq rad R$ by $(4.11)$ ; and $rad R \subseteq 𝔪$ because $𝔪$ is a maximal ideal with $R / 𝔪$ a field. So $rad R = 𝔪$ and $(19.1) (3)$ applies.

Example(19.10)Group algebras of finite

p

-groups

Let $k$ be a field of characteristic $p > 0$ and $G$ a finite $p$ -group. Then $rad (k G)$ is the augmentation ideal $𝔞 = \ker (k G \to k)$ , with $𝔞^{| G |} = 0$ , and $k G / 𝔞 ≅ k$ . So $k G$ is an artinian local ring, indeed completely primary.

Proposition(19.11)Group algebras over a local coefficient ring

Let $(R, 𝔪)$ be a commutative local ring whose residue field $k = R / 𝔪$ has characteristic $p > 0$ , and let $G$ be a **finite $p$ -group**. Then $A = R G$ is a local ring with $A / rad A ≅ k$ , and $rad A$ is the ideal $I$ generated by $𝔪$ together with all $g - 1$ for $g \in G$ — the kernel of the reduced augmentation map $R G \to R \to k$ .

Proof

Let $V$ be any simple left $A$ -module. Being simple, $V$ is cyclic over $A$ ; since $A$ is generated as an $R$ -module by the finitely many elements of $G$ , $V$ is a finitely generated $R$ -module. Nakayama's Lemma $(4.22)$ then gives $𝔪 V \neq V$ .

Because $R$ is commutative and central in $R G$ , the subset $𝔪 V$ is an $A$ -submodule of $V$ . Simplicity and $𝔪 V \neq V$ force $𝔪 V = 0$ , so $V$ is a module over $A / 𝔪 A ≅ k G$ , and it is simple as such.

Now $(8.8)$ applies: for a finite $p$ -group $G$ over a field of characteristic $p$ , the only simple $k G$ -module is the trivial one, so $G$ acts trivially on $V$ , i.e. $(g - 1) V = 0$ for all $g \in G$ .

Thus $I$ annihilates every simple left $A$ -module, giving $I \subseteq rad A$ . On the other hand $A / I ≅ k$ : killing all $g - 1$ collapses $R G$ onto $R$ , and killing $𝔪$ then gives $k$ . Hence $I$ is a maximal ideal, and since $rad A$ is proper we get $rad A = I$ and $A / rad A ≅ k$ , a field. By $(19.1) (3)$ , $A$ is local.

Remark(19.11′)Dropping commutativity of the coefficients

The proposition remains true for $(R, 𝔪)$ a noncommutative local ring with residue division ring of characteristic $p$ . The reduction is the same: Nakayama kills $𝔪 V$ , leaving a simple module over $k G$ for $k$ a division ring of characteristic $p$ , and the argument of $(8.8)$ still shows that a finite $p$ -group acts trivially. This form of the result is what is used in the integral representation theory of finite groups, where $R$ is typically a complete discrete valuation ring.

Remark—The converse for group algebras

For a field $k$ and a nontrivial finite group $G$ , $k G$ is local only in the situation of $(19.10)$ : $k G$ local is equivalent to $char k = p > 0$ and $G$ a $p$ -group, and equally to $k G$ being indecomposable as a module over itself. That converse is developed in When Is a Group Algebra a Local Ring?

Proof Techniques and Method

How these verifications work, and which move to reuse.

Move 1

Guess the ideal from the grading

If the ring is graded or filtered with degree-zero part a division ring, the positive part is the only candidate for the radical — and its nilpotence is immediate from the grading.

Move 2

Nil goes down, maximal comes back

$(4.11)$ gives $𝔪 \subseteq rad R$ when $𝔪$ is nil; maximality of $𝔪$ gives the reverse. Two one-line inclusions replace any direct computation of the radical.

Move 3

Compute units by a convergent series

In power series and other complete settings, $1 - g$ is inverted by $\sum g^{i}$ whenever the powers of $g$ shrink. This is what makes non-nilpotent radicals manageable.

Move 4

Nakayama to strip the coefficients

Over $R G$ with $R$ local, a simple module is finitely generated over $R$ , so Nakayama forces $𝔪 V = 0$ and reduces the question to the residue field. This is the whole content of $(19.11)$ .

Move 5

$p$ -groups act trivially

In characteristic $p$ a finite $p$ -group has only the trivial simple module. The reason is that $g - 1$ is nilpotent for $g$ of $p$ -power order: $(g - 1)^{p^{e}} = g^{p^{e}} - 1 = 0$ .

Move 6

Transport along a quotient

If $𝔘 \subseteq rad A$ then $rad (A / 𝔘) = (rad A) / 𝔘$ by $(4.6)$ . Identify the quotient as something with zero radical and the radical is pinned down exactly.

Move 5 is worth spelling out because it is where characteristic $p$ enters. Over $ℂ$ , $g - 1$ for $g$ of order $p$ is a semisimple element with $p$ distinct eigenvalues, and $ℂ G$ is semisimple by Maschke's theorem — the opposite extreme. Locality of $k G$ is a purely modular phenomenon.

Worked Example

A: the exterior algebra on two generators

Take $k = ℚ$ , $V = k e_{1} \oplus k e_{2}$ , and $R = Λ (V)$ , a $4$ -dimensional $ℚ$ -algebra with basis $1, e_{1}, e_{2}, e_{1} e_{2}$ and relations $e_{1}^{2} = e_{2}^{2} = 0$ , $e_{2} e_{1} = - e_{1} e_{2}$ .

𝔪 = ℚ e_{1} \oplus ℚ e_{2} \oplus ℚ e_{1} e_{2}, 𝔪^{2} = ℚ e_{1} e_{2}, 𝔪^{3} = 0 .

(E.1)

Dimensions $3, 1, 0$ — the Loewy series of a completely primary algebra of Loewy length $3$ .

Check $𝔪^{2}$ by hand: the products of degree-one basis elements are $e_{1} e_{1} = 0$ , $e_{2} e_{2} = 0$ , $e_{1} e_{2}$ , and $e_{2} e_{1} = - e_{1} e_{2}$ . So $𝔪^{2}$ is spanned by $e_{1} e_{2}$ alone, of dimension $1$ . Multiplying once more lands in degree $\geq 3 > \dim V$ , which is zero.

Units: an element $u = a + w$ with $a \in ℚ^{\times}$ and $w \in 𝔪$ is invertible, with $u^{- 1} = a^{- 1} - a^{- 1} w a^{- 1} + a^{- 1} w a^{- 1} w a^{- 1}$ , the series terminating because $𝔪^{3} = 0$ . Concretely $(1 + e_{1})^{- 1} = 1 - e_{1}$ , since $e_{1}^{2} = 0$ ; and $(1 + e_{1} + e_{2})^{- 1} = 1 - e_{1} - e_{2}$ , since $(e_{1} + e_{2})^{2} = e_{1} e_{2} + e_{2} e_{1} = 0$ .

So $R ∖ U (R) = 𝔪$ , an ideal, and $R$ is local by $(19.1) (4)$ with residue field $ℚ$ . It is noncommutative because $e_{1} e_{2} = - e_{2} e_{1} \neq e_{2} e_{1}$ .

B: a group algebra with non-nilpotent radical

Take $R = ℤ_{(2)} = {a / b \in ℚ : b odd}$ , a commutative local ring with $𝔪 = 2 ℤ_{(2)}$ and residue field $k = 𝔽_{2}$ of characteristic $2$ . Let $G = {1, g}$ be cyclic of order $2$ — a $2$ -group. By $(19.11)$ , $A = R G$ is local.

Make it explicit. Put $t = g - 1$ . Then $t^{2} = g^{2} - 2 g + 1 = 2 - 2 g = - 2 t$ , so

A ≅ ℤ_{(2)} [t] / (t^{2} + 2 t), rad A = (2, t), A / rad A ≅ 𝔽_{2} .

(E.2)

The two halves of the example bracket the phenomenon: $(19.9)$ and $(19.10)$ produce completely primary artinian rings, while $(19.7)$ and $(19.11)$ over an infinite coefficient ring produce local rings with no chain condition at all.

Frameworks and Models

The examples organise by how the division-ring quotient is produced.

Sources of local rings
- Degenerate — the radical is zero
  - division rings, including fields
  - the real quaternions $ℍ$
  - cyclic algebras that happen to be division algebras
- Complete / filtered — the radical is not nil but powers shrink
  - $R [[x]]$ over a local $R$
  - twisted power series $R [[x; σ]]$
  - $ℤ_{p}$ and other discrete valuation rings
  - valuation rings of a division ring $(19.3) (b)$
- Graded / nilpotent — the radical is nilpotent — completely primary
  - exterior algebras $Λ (V)$
  - constant-diagonal triangular matrices $(19.8)$
  - $k G$ for $G$ a finite $p$ -group, $char k = p$
  - truncated polynomial and tensor algebras
- Endomorphism rings — locality is a theorem, not a construction
  - $End (M_{R})$ for $M$ indecomposable of finite length
  - $End (M_{R})$ for $M$ indecomposable injective

The fourth branch is the one with no commutative shadow, and it is the reason local rings matter here at all: it supplies the hypothesis of the Krull–Schmidt–Azumaya theorem. See Strongly Indecomposable Modules and Local Endomorphism Rings.

Comparison and Classification

The standard examples, side by side
Ring	$rad$	Residue object	$rad$ nilpotent?	Commutative?
Division ring $D$	$0$	$D$	yes (trivially)	only if $D$ is a field
$ℤ_{p}$	$p ℤ_{p}$	$𝔽_{p}$	no, not even nil	yes
$k [[x]]$	$(x)$	$k$	no, not even nil	yes
$k [[x; σ]]$ , $σ \neq id$	$(x)$	$k$	no, not even nil	no
$A = k \cdot 1 \oplus J \subseteq T_{n} (k)$	$J$	$k$	yes, index $n$	no for $n \geq 3$
$Λ (V)$ , $\dim V = n$	$Λ^{\geq 1} (V)$	$k$	yes, index $n + 1$	no for $n \geq 2$ , $char k \neq 2$
$k G$ , $G$ a $p$ -group, $char k = p$	augmentation ideal	$k$	yes, index $\leq \| G \|$	only if $G$ is abelian
$ℤ_{(p)} G$ , $G$ a $p$ -group	$(p, g - 1)$	$𝔽_{p}$	no	only if $G$ is abelian

Which additional properties each family enjoys
	Artinian	Noetherian	Domain	Completely primary
Division ring	yes	yes	yes	yes
$k [[x; σ]]$	no	yes	yes	no
$ℤ_{p}$	no	yes	yes	no
$Λ (V)$ , $\dim V \geq 1$	yes	yes	no	yes
$k G$ , $G$ a nontrivial $p$ -group	yes	yes	no	yes
Valuation ring of infinite rank	no	no	yes	no

Which additional properties each family enjoys

Relationship Map

The constructions interlock: each preserves or creates locality in a predictable way.

k

a field

k [[x; σ]]

local, noncommutative

k [[x; σ]] [[y; τ]]

localiterate

Quotients. $R$ local and $I ⊊ R$ an ideal $\Rightarrow$ $R / I$ local. This is why $𝔽_{2} G$ inherits locality from $ℤ_{(2)} G$ .
Power series. $R$ local $\Rightarrow$ $R [[x; σ]]$ local, with the same residue division ring. Polynomial rings do not behave this way: $k [x]$ has infinitely many maximal ideals.
Matrices. $R$ local $\Rightarrow$ $M_{n} (R)$ is not local for $n \geq 2$ ; instead it is semilocal, with semisimple quotient $M_{n} (R / rad R)$ .
Group algebras. $R$ local with residue characteristic $p$ and $G$ a finite $p$ -group $\Rightarrow$ $R G$ local; for any other finite $G$ the conclusion fails.
Centres. $R$ local $\Rightarrow$ $Z (R)$ is a commutative local ring, so every noncommutative example above has a commutative local ring hiding inside it.

Applications and Industry Use

Applications here means where this structure is used — inside mathematics and in the engineering and computing disciplines that consume it.

Modular representation theory

Blocks and vertices

$k G$ for $G$ a $p$ -group being local is the base case of block theory: every block of $k H$ for general $H$ is analysed by restriction to $p$ -subgroups, where the group algebra is completely primary.

Integral representations

Coefficient rings

$(19.11)$ over a complete discrete valuation ring $R$ with residue characteristic $p$ underlies the reduction-mod- $p$ machinery relating characteristic-zero and modular representations.

Coding theory

Galois and chain rings

Codes over $ℤ / p^{n}$ and Galois rings live over finite chain rings, which are exactly the finite local rings with principal maximal ideal; the residue field carries the associated code.

Physics and geometry

Exterior algebras

$Λ (V)$ is the algebra of differential forms at a point and the fermionic side of a supersymmetric algebra. Its locality is the statement that a form is invertible iff its scalar part is nonzero.

Control and signals

Skew series

Twisted power series $k [[x; σ]]$ model linear time-varying systems where the shift operator does not commute with the coefficients; invertibility of a transfer element reduces to its constant term.

Number theory

Completions

$ℤ_{p}$ and, more generally, valuation rings inside division algebras over local fields are the arithmetic prototypes; the noncommutative case classifies division algebras over $p$ -adic fields.

Computational Notes

Computational notes cover algorithms, cost and library behaviour rather than manufacturing process.

For a finite-dimensional algebra $A$ over a field given by structure constants, testing locality is one radical computation plus a check that $\dim_{k} A / rad A$ equals the dimension of a single division algebra factor. In characteristic $0$ the radical is the kernel of the trace form, an $O (n^{3})$ nullspace computation for $n = \dim_{k} A$ .
In characteristic $p$ the trace form is inadequate — the Friedl–Rónyai algorithm handles it in polynomial time. This matters exactly for the group algebras of $(19.10)$ , where the trace form is identically degenerate.
For $k G$ with $G$ a $p$ -group in characteristic $p$ , no computation is needed: locality is automatic and the radical is the augmentation ideal, of dimension $| G | - 1$ , computable as the kernel of a single linear map.
Power series rings are not finitely presented; software handles them by truncation. Deciding whether a truncated series is a unit is one test on the constant term, and the inverse to precision $N$ costs $O (N^{2})$ coefficient multiplications by the geometric-series recursion, or $O (N \log N)$ by Newton iteration in the untwisted commutative case.
Exterior algebras are supported natively in GAP, Macaulay2 and Singular; their radical is the ideal of positive-degree elements and needs no algorithm at all.

Failure Modes and Common Mistakes

Do not take the whole upper triangular ring $T_{n} (k)$ for $n \geq 2$ to be local: its diagonal idempotents forbid it. Only the constant-diagonal subring $A$ of $(19.8)$ is local.
Do not assume $Λ (V)$ is noncommutative: in characteristic $2$ , or when $\dim V \leq 1$ , it is commutative. Locality holds regardless.
Do not extend $(19.11)$ to a coefficient ring of residue characteristic $\neq p$ : with $| G |$ invertible in $R$ , Maschke's argument applies and $R G$ acquires idempotents.
Do not confuse the augmentation ideal with the radical in general. They coincide for $p$ -groups in characteristic $p$ ; for $ℂ S_{3}$ the augmentation ideal is a direct summand and the radical is zero.

Quick Reference

$R [[x; σ]]$

rad = {f : f (0) \in rad R}

; local iff

R

$T_{n} (k)$ constant diagonal

rad =

strictly upper triangular,

J^{n} = 0

$Λ (V)$ , $\dim V = n$

rad = Λ^{\geq 1} (V)

𝔪^{n + 1} = 0

$k G$ , $G$ a $p$ -group

rad =

augmentation ideal,

𝔞^{| G |} = 0

$R G$ , $(R, 𝔪)$ locallocal if

char (R / 𝔪) = p

and

G

is a

p

-group

Unit test in $R [[x]]$

f \in U

iff

f (0) \in U (R)

Valuation criterion

R \subseteq D

with

d

d^{- 1}

R

\Rightarrow

R

local

Never local

M_{n} (R)

for

n \geq 2

;

T_{n} (k)

for

n \geq 2

;

ℂ G

for

G \neq 1

Where each example is proved
Example	Reference	Radical nilpotent?
Localisation $R_{𝔭}$	(19.4)	no in general
Local ring of a variety	(19.5)	no
Germs on a Riemann surface	(19.6)	no
Valuation rings, $ℤ_{p}$	(19.7)	no
Constant-diagonal triangular	(19.8)	yes
Exterior algebra	(19.9)	yes
$k G$ , $G$ a $p$ -group	(19.10)	yes
$R G$ over a local ring	(19.11)	only if $rad R$ is

Frequently Asked Questions

Why is there no noncommutative localisation producing local rings on demand?

Forming a ring of fractions with respect to a multiplicative set $S$ requires the Ore condition — every pair $(r, s) \in R \times S$ must satisfy $r s' = s r'$ for some $r' \in R$ , $s' \in S$ . For a general ring, and in particular for the complement of a prime ideal, that condition fails, and the universal construction that ignores it can collapse. So the localisation route to local rings simply is not available in general, and the noncommutative examples come from completions, gradings and endomorphism rings instead.

Is a twisted power series ring

k [[x; σ]]

ever commutative?

Exactly when $σ$ is the identity. The relation $x a = σ (a) x$ shows that $x$ commutes with $a$ iff $σ (a) = a$ , so nontrivial $σ$ gives a genuinely noncommutative ring. It is always a local domain with maximal ideal $(x)$ and residue division ring $k$ , so this is the cheapest way to produce a noncommutative local ring.

Why does the exterior algebra fail to be a domain even though it is local?

Because $e \land e = 0$ for every $e \in V$ , so every degree-one element is a nonzero square-zero element. Locality says nothing about zero divisors: the two conditions are independent. $k [[x]]$ is a local domain, $Λ (V)$ is a local ring with plenty of zero divisors, and $ℤ$ is a domain that is not local.

How large can the nilpotency index of

rad (k G)

be for a

p

-group?

Lam's bound is $𝔞^{| G |} = 0$ , which is generally not sharp. For $G$ cyclic of order $p^{n}$ the augmentation ideal is generated by $t = g - 1$ with $t^{p^{n}} = 0$ and $t^{p^{n} - 1} \neq 0$ , so the index is exactly $| G |$ there. For non-cyclic $p$ -groups the index is strictly smaller: for the Klein four group in characteristic $2$ the augmentation ideal has cube zero, well below $| G | = 4$ .

Does

(19.11)

need the group to be finite?

Yes, in two places. Finiteness makes $A = R G$ a finitely generated $R$ -module, which is what lets Nakayama's Lemma apply to a cyclic $A$ -module; and the triviality of the $G$ -action on simple modules in $(8.8)$ is proved for finite $p$ -groups. For infinite locally finite $p$ -groups the augmentation ideal is nil but not nilpotent, and the conclusion has to be argued differently.

Which of these examples are artinian?

Exactly the completely primary ones with finite dimension: the constant-diagonal triangular ring, the exterior algebra, and $k G$ for $G$ a finite $p$ -group over a field. Division rings are trivially artinian. The power series rings, $ℤ_{p}$ , valuation rings and $ℤ_{(p)} G$ are not — their radicals are not nil.

References

T. Y. Lam, A First Course in Noncommutative Rings, Graduate Texts in Mathematics 131, Springer-Verlag, 1991, §19 (pp. 293–310), especially (19.4)–(19.11).
D. S. Passman, The Algebraic Structure of Group Rings, Wiley-Interscience, 1977, Chapters 1 and 3.
C. W. Curtis and I. Reiner, Methods of Representation Theory, Volume I, Wiley-Interscience, 1981, §6.
N. Jacobson, Structure of Rings, American Mathematical Society Colloquium Publications 37, revised edition, 1964.
O. F. G. Schilling, The Theory of Valuations, Mathematical Surveys 4, American Mathematical Society, 1950.
J.-P. Serre, Local Fields, Graduate Texts in Mathematics 67, Springer-Verlag, 1979, Chapters I–II.

AI Suggested Questions

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01Executive Summary

02Overview

03Learning Objectives

04Core Concepts

The commutative stock, for orientation

Power series: the first noncommutative supply

Graded algebras with one-dimensional degree zero

05Key Results

06Proof Techniques and Method

Guess the ideal from the grading

Nil goes down, maximal comes back

Compute units by a convergent series

Nakayama to strip the coefficients

p-groups act trivially

Transport along a quotient

07Worked Example

A: the exterior algebra on two generators

B: a group algebra with non-nilpotent radical

08Frameworks and Models

09Comparison and Classification

10Relationship Map

11Applications and Industry Use

Blocks and vertices

Coefficient rings

Galois and chain rings

Exterior algebras

Skew series

Completions

12Computational Notes

13Failure Modes and Common Mistakes

14Quick Reference

15Frequently Asked Questions

16Related KEVOS Topics

17References

18AI Suggested Questions

Continue learning

Executive Summary

Overview

Learning Objectives

Core Concepts

Key Results

Proof Techniques and Method

$p$ -groups act trivially

Worked Example

Frameworks and Models

Comparison and Classification

Relationship Map

Applications and Industry Use

Computational Notes

Failure Modes and Common Mistakes

Quick Reference

Frequently Asked Questions

Related KEVOS Topics

References

AI Suggested Questions