The Radical of Matrix Rings and Power Series Rings

Q: Why is the matrix formula entrywise while the power series formula constrains only the constant term?

Because the mechanisms differ. In M_n(R) the matrix units conjugate any entry into the (1,1) position, so a two-sided ideal cannot contain a matrix without containing all matrices with those entries — membership is forced entry by entry. In R[[t]] nothing moves coefficients around; instead the whole ideal tR[[t]] is swallowed by the radical because 1 - tg is invertible by a convergent geometric series. The radical is then simply the preimage of rad R.

Q: Is the Jacobson radical a Morita invariant?

Yes. rad M_n(R) = M_n(rad R) together with the corner ring theorem rad(eRe) = e(rad R)e for a full idempotent e gives invariance under the generators of Morita equivalence. Concretely, R and M_n(R) are semiprimitive together, semilocal together and semiperfect together, and their radical quotients correspond.

Q: Does R[[t]] inherit chain conditions from R?

Not the descending one. For R ≠ 0 the element t lies in rad R[[t]] and is not nilpotent, so the radical is not nilpotent and R[[t]] cannot be left artinian, by (4.12) — even when R is a field. This is the standard reminder that a local ring need not be an artinian local ring.

Q: What is the analogue for infinite matrix rings?

There is none of this form. Both proofs given here use n < ∈fty: the row reduction terminates after n pivots, and the ideal correspondence for M_n(R) is a finite-matrix statement. Rings of row-finite or column-finite infinite matrices need separate analysis, and the naive entrywise guess should not be assumed.

Q: Why does the polynomial ring behave so much worse than the power series ring?

Degrees add in R[t], so 1 - tg is essentially never a unit and no analogue of tR[[t]] is available inside the radical. Amitsur's theorem (5.10) still gives rad R[t] = N[t] with N = R rad R[t] nil, but deciding whether N is the upper nilradical Nil^*R is Problem (5.12), equivalent to Köthe's conjecture.

Q: Does the skew case need to be an automorphism?

The proof as given does. Bijectivity of is what makes xA = Ax a two-sided ideal with quotient R; for a merely injective endomorphism the left and right ideals generated by x can differ, and the clean preimage description is not justified without further hypotheses.

Executive Summary

Most change-of-ring questions about $rad$ are hard. Two are not. For a full matrix ring the radical is computed entry by entry, $rad M_{n} (R) = M_{n} (rad R)$ ; for a formal power series ring it is the preimage of $rad R$ under evaluation at $t = 0$ , so $rad R [[t]] = rad R + t R [[t]]$ . Both answers hold for an arbitrary ring with identity, with no chain condition and no commutativity.

The two results come from different mechanisms — matrix units on one side, convergent geometric series on the other — and it is worth seeing both, because between them they explain why the polynomial ring $R [t]$ resists description and why that resistance is equivalent to a famous open problem.

$M_{n} (rad R)$

rad M_{n} (R)

$rad R + t R [[t]]$

rad R [[t]]

Ex. 5.6Lam's power series exercise

NoneHypotheses on

R

Overview

Section 5 of Lam asks how $rad$ moves between a ring and an extension of it. The general results — the Behaviour of the Radical under Ring Extensions page — give one-way inclusions under hypotheses such as * $R$ is an $R$ -module direct summand of $S$ * or * $S$ is generated over $R$ by elements centralising $R$ *. The two constructions on this page are the cases where those inclusions can be sharpened all the way to equalities and stated in closed form.

rad M_{n} (R) = M_{n} (rad R), rad R [[t]] = {f \in R [[t]] : f (0) \in rad R}

(5.M)

The two closed formulas of this page. The first is Lam's Example 7 in §4; the second is Exercise 5.6.

The first formula is the reason $rad$ is a Morita invariant, which in turn is why Wedderburn–Artin theory can be applied after passing to $R / rad R$ without worrying about matrix size. The second is the reason power series rings are the standard supply of noncommutative local rings, as used on the Local Rings page.

The contrast with $R [t]$ is instructive and is not a defect of exposition: Amitsur's theorem says $rad R [t] = N [t]$ for a nil ideal $N \subseteq R$ , but identifying $N$ with the upper nilradical ${Nil}^{*} R$ is equivalent to Köthe's conjecture.

Learning Objectives

Prove $rad M_{n} (R) = M_{n} (rad R)$ for an arbitrary ring $R$ and every $n \geq 1$ .
Deduce $M_{n} (R) / rad M_{n} (R) ≅ M_{n} (R / rad R)$ and the Morita invariance of the radical.
Prove that $f \in R [[t]]$ is a unit exactly when $f (0) \in U (R)$ , and derive $rad R [[t]] = rad R + t R [[t]]$ .
Extend the power series computation to the skew ring $R [[x; σ]]$ for $σ \in Aut (R)$ .
Compute $rad$ for upper triangular and generalised triangular matrix rings.
Explain why $R [[t]]$ is never left artinian and $M_{n} (R)$ is never local for $n \geq 2$ .

Definitions

Definition(5.M.1)The two constructions

For a ring $R$ with identity and $n \geq 1$ , $M_{n} (R)$ denotes the ring of $n \times n$ matrices over $R$ , with matrix units $E_{i j}$ satisfying $E_{i j} E_{k ℓ} = δ_{j k} E_{i ℓ}$ . For a two-sided ideal $I \subseteq R$ , $M_{n} (I)$ denotes the set of matrices all of whose entries lie in $I$ ; it is a two-sided ideal of $M_{n} (R)$ .

$R [[t]]$ denotes the ring of formal power series $f = \sum_{i \geq 0} a_{i} t^{i}$ with $a_{i} \in R$ and $t$ central. Evaluation at zero, ${ev}_{0} (f) = a_{0}$ , is a surjective ring homomorphism $R [[t]] \to R$ with kernel $t R [[t]]$ .

$M_{n} (I)$ versus $I \cdot M_{n} (R)$: These coincide for a two-sided ideal $I$ : every matrix with entries in $I$ is a finite sum $\sum a_{i j} E_{i j}$ with $a_{i j} \in I$ .
${ev}_{0}$: The constant-term homomorphism $R [[t]] \to R$ . Its kernel $t R [[t]]$ is the ideal of series with zero constant term.
$R [[x; σ]]$: Formal power series $\sum_{i \geq 0} a_{i} x^{i}$ with the rule $x r = σ (r) x$ for an automorphism $σ$ of $R$ . Multiplication is well defined because each coefficient of a product is a finite sum.
$T_{n} (R)$: The subring of $M_{n} (R)$ of upper triangular matrices. Its strictly upper triangular part is a nilpotent ideal of index $n$ .
Semiprimitive: $rad R = 0$ ; also called Jacobson semisimple or J-semisimple.

All rings have an identity, all modules are unital, and $n$ is a finite positive integer throughout. The finiteness of $n$ is used in both proofs of the matrix theorem.

Core Concepts

Two mechanisms for transporting the radical

It helps to separate the two arguments before reading either proof, because they generalise in different directions.

Mechanism A

Matrix units move entries

Because $E_{1 i} A E_{j 1} = a_{i j} E_{11}$ , membership of a matrix in a two-sided ideal is equivalent to membership of each of its entries in the corresponding ideal of $R$ . The radical, being a two-sided ideal, is therefore detected one entry at a time.

Mechanism B

Geometric series converge

In $R [[t]]$ the sum $\sum_{i \geq 0} (t g)^{i}$ makes sense because only finitely many terms contribute to each coefficient. So $1 - t g$ is a unit for every $g$ , and $t R [[t]]$ is forced inside the radical by the maximality property $(4.5)$ .

Why the second mechanism gives a preimage

Once an ideal $𝔄$ is known to lie inside $rad A$ , the quotient rule $(4.6)$ applies: $rad (A / 𝔄) = (rad A) / 𝔄$ . Taking $A = R [[t]]$ and $𝔄 = t A$ , whose quotient is $R$ , converts the problem into a computation in $R$ and returns the answer as a preimage.

1 + t A \subseteq U (A)

t A \subseteq rad A

rad (A / t A) = (rad A) / t A

rad A = {ev}_{0}^{- 1} (rad R)

This is a template, not a one-off. Any surjection $A ↠ R$ whose kernel consists of quasi-regular elements and is an ideal computes $rad A$ from $rad R$ in exactly the same way — nilpotent kernels, T-nilpotent kernels and complete filtrations all qualify.

Key Results

Theorem§4, Example 7Radical of a full matrix ring

Let $R$ be any ring with identity and let $n \geq 1$ . Then

rad M_{n} (R) = M_{n} (rad R) .

That is, a matrix lies in the radical of $M_{n} (R)$ if and only if each of its $n^{2}$ entries lies in $rad R$ .

Proof

Write $J = rad R$ .

** $M_{n} (J) \subseteq rad M_{n} (R)$ .** Since $J$ is a two-sided ideal of $R$ , $M_{n} (J)$ is a two-sided ideal of $M_{n} (R)$ . By the maximality property $(4.5)$ it is enough to show that $I_{n} + A \in U (M_{n} (R))$ for every $A \in M_{n} (J)$ . Induct on $n$ . For $n = 1$ this is $(4.5)$ in $R$ . For $n > 1$ put $U = I_{n} + A = (u_{i j})$ , so $u_{11} = 1 + a_{11} \in U (R)$ and $u_{i 1} = a_{i 1} \in J$ for $i \geq 2$ . Left-multiplying $U$ by the invertible matrices $I_{n} - u_{i 1} u_{11}^{- 1} E_{i 1}$ for $i = 2, \dots, n$ clears the first column below the diagonal and produces

U' = (\begin{matrix} u_{11} & v \\ 0 & W \end{matrix}), W = (u_{i j} - u_{i 1} u_{11}^{- 1} u_{1 j})_{2 \leq i, j \leq n} .

For $i \geq 2$ the correction term $u_{i 1} u_{11}^{- 1} u_{1 j}$ lies in $J$ , because $u_{i 1} \in J$ and $J$ is an ideal. Hence $W = I_{n - 1} + A'$ with $A' \in M_{n - 1} (J)$ , so $W \in U (M_{n - 1} (R))$ by the inductive hypothesis. A block upper triangular matrix with invertible diagonal blocks is invertible — explicitly, the inverse of $U'$ is $(\begin{matrix} u_{11}^{- 1} & - u_{11}^{- 1} v W^{- 1} \\ 0 & W^{- 1} \end{matrix})$ — and $U$ is a product of invertible matrices with $U'$ , hence invertible.

** $rad M_{n} (R) \subseteq M_{n} (J)$ .** Let $A = (a_{i j}) \in rad M_{n} (R)$ and fix indices $i, j$ . The radical is a two-sided ideal, so $E_{1 i} A E_{j 1} = a_{i j} E_{11}$ also lies in it. Fix $b \in R$ and set $d = b a_{i j}$ . Applying $(4.1)$ inside $M_{n} (R)$ with the element $b E_{11}$ , the matrix $I_{n} - (b E_{11}) (a_{i j} E_{11}) = I_{n} - d E_{11}$ is left-invertible: there is $C = (c_{k ℓ})$ with $C (I_{n} - d E_{11}) = I_{n}$ . Comparing $(1, 1)$ entries gives $c_{11} - c_{11} d = c_{11} (1 - d) = 1$ , so $1 - b a_{i j}$ is left-invertible in $R$ . As $b$ was arbitrary, $(4.1)$ gives $a_{i j} \in J$ .

The two inclusions give the stated equality.

Corollary(5.M.2)Consequences for matrix rings

Let $R$ be a ring and $n \geq 1$ . Then:

$M_{n} (R) / rad M_{n} (R) ≅ M_{n} (R / rad R)$ ;
$M_{n} (R)$ is semiprimitive if and only if $R$ is semiprimitive;
$(rad M_{n} (R))^{m} = M_{n} ((rad R)^{m})$ for every $m \geq 1$ , so $rad M_{n} (R)$ is nilpotent of index $m$ exactly when $rad R$ is;
for $R \neq 0$ and $n \geq 2$ , $M_{n} (R)$ is never a local ring.

Proof

(1) Entrywise reduction $M_{n} (R) \to M_{n} (R / rad R)$ is a surjective ring homomorphism with kernel $M_{n} (rad R) = rad M_{n} (R)$ . (2) is immediate since $M_{n} (I) = 0$ forces $I = 0$ . (3) follows from $M_{n} (I) M_{n} (I') = M_{n} (I I')$ for two-sided ideals $I, I'$ , which holds because $E_{i k}$ can be used to place any product $a a'$ in any position. (4) A ring is local exactly when its quotient by the radical is a division ring; by (1) that quotient is $M_{n} (R / rad R)$ , and for $n \geq 2$ this contains the nonzero zero-divisors $E_{11}, E_{22}$ with $E_{11} E_{22} = 0$ .

Lemma(5.M.3)Units of a power series ring

For any ring $R$ and $f \in R [[t]]$ : $f \in U (R [[t]])$ if and only if $f (0) \in U (R)$ .

Proof

If $f$ is a unit then so is its image under the ring homomorphism ${ev}_{0}$ . Conversely let $a_{0} = f (0) \in U (R)$ and write $f = a_{0} + t h$ with $h \in R [[t]]$ . Since $t$ is central, $f = a_{0} (1 + t g)$ with $g = a_{0}^{- 1} h$ . The series $\sum_{i \geq 0} (- t g)^{i}$ is a well-defined element of $R [[t]]$ , since the coefficient of $t^{m}$ receives contributions only from the terms with $i \leq m$ , and it is a two-sided inverse of $1 + t g$ . Hence $f \in U (R [[t]])$ .

TheoremEx. 5.6Radical of a formal power series ring

Let $R$ be any ring with identity and $A = R [[t]]$ with $t$ a central indeterminate. Then

rad A = {ev}_{0}^{- 1} (rad R) = {a + t f (t) : a \in rad R, f \in A} = rad R + t A .

In particular $A \cap rad A = rad R$ under the identification of $R$ with the constant series, and $A / rad A ≅ R / rad R$ .

Proof

The ideal $t A$ is two-sided, and $1 + t A \subseteq U (A)$ by $(5 . M . 3)$ since every element of $1 + t A$ has constant term $1$ . By the maximality property $(4.5)$ , $t A \subseteq rad A$ .

Because $t A$ is an ideal contained in $rad A$ , the quotient rule $(4.6)$ gives $rad (A / t A) = (rad A) / t A$ . The isomorphism $A / t A ≅ R$ induced by ${ev}_{0}$ identifies the left-hand side with $rad R$ . Therefore $(rad A) / t A$ corresponds to $rad R$ , i.e. $rad A = {ev}_{0}^{- 1} (rad R) = rad R + t A$ . The two final statements follow by restricting ${ev}_{0}$ to constants and by the first isomorphism theorem.

Corollary(5.M.4)Consequences for power series rings

Let $R \neq 0$ be a ring and $A = R [[t]]$ . Then:

$A$ is never semiprimitive, since $0 \neq t \in rad A$ ;
$rad A$ is never nil, since $t$ is not nilpotent; consequently $A$ is never left artinian, by $(4.12)$ ;
$A$ is local if and only if $R$ is local, and then the residue division rings agree: $A / rad A ≅ R / rad R$ ;
$A$ is semilocal if and only if $R$ is semilocal.

Proposition(5.M.5)Skew power series

Let $R$ be a ring, $σ$ an automorphism of $R$ , and $A = R [[x; σ]]$ the skew power series ring with $x r = σ (r) x$ . Then $x A = A x$ is a two-sided ideal, $A / x A ≅ R$ , and

rad R [[x; σ]] = rad R + x A .

The proof is that of the untwisted case verbatim: $r x = x σ^{- 1} (r)$ shows $A x \subseteq x A$ , geometric series still converge $x$ -adically, and $rad R$ is invariant under every automorphism of $R$ , so the constant-term description is unambiguous.

Proposition(5.M.6)Triangular matrix rings

Let $R$ be a ring and $T_{n} (R) \subseteq M_{n} (R)$ the ring of upper triangular matrices. Then

rad T_{n} (R) = {(a_{i j}) \in T_{n} (R) : a_{i i} \in rad R for 1 \leq i \leq n} .

More generally, if $A$ and $B$ are rings and $M$ is an $(A, B)$ -bimodule, the generalised triangular ring $(\begin{matrix} A & M \\ 0 & B \end{matrix})$ has radical $(\begin{matrix} rad A & M \\ 0 & rad B \end{matrix})$ .

Proof

In the two-block case, $𝔑 = (\begin{matrix} 0 & M \\ 0 & 0 \end{matrix})$ is a two-sided ideal with $𝔑^{2} = 0$ , hence $𝔑 \subseteq rad$ by $(4.11)$ . The quotient by $𝔑$ is $A \times B$ , whose radical is $rad A \times rad B$ because units in a product are componentwise units. Now $(4.6)$ identifies the radical with the preimage, which is the stated ideal. The $T_{n} (R)$ statement is the same argument with $𝔑$ the strictly upper triangular matrices, an ideal with $𝔑^{n} = 0$ and quotient $R \times \dots \times R$ ( $n$ factors).

Remark(21.10), (21.14)The structural explanation

For any idempotent $e \in R$ the corner ring theorem states $rad (e R e) = e R e \cap rad R = e (rad R) e$ . Taking $R = M_{n} (S)$ and $e = E_{11}$ gives $e R e ≅ S$ , and $e$ is a full idempotent, so the ideal correspondence for $e R e$ matches $rad M_{n} (S)$ with $rad S$ . This is Lam's second derivation of the matrix formula, and it is the one that generalises: it is really a statement about Morita equivalence, developed on the Corner Rings page.

Proof Techniques and Method

How these proofs work, and which move to reuse elsewhere.

Move 1

Squeeze with $(4.5)$ then $(4.6)$

To compute $rad A$ , find an ideal $𝔄$ with $1 + 𝔄 \subseteq U (A)$ and a recognisable quotient $A / 𝔄$ . Then $𝔄 \subseteq rad A$ by $(4.5)$ and $rad A$ is the preimage of $rad (A / 𝔄)$ by $(4.6)$ .

Move 2

Conjugate by matrix units

$E_{1 i} A E_{j 1} = a_{i j} E_{11}$ turns any statement about matrices in a two-sided ideal into a statement about single entries. Combined with $(4.1)$ this needs only left-invertibility, so no Dedekind-finiteness is assumed.

Move 3

Row reduce inside the radical

Gaussian elimination works over any ring provided the pivots are units. Entries of $I_{n} + A$ with $A$ in the radical give pivots in $1 + rad R$ , which are units by $(4.5)$ , and the corrections stay in the radical because it is an ideal.

Move 1 is the reusable one. It computes the radical of $R [[t]]$ , of $T_{n} (R)$ , of $R [t] / (t^{m})$ , of any ring complete with respect to a filtration, and of any surjection with nilpotent or T-nilpotent kernel. When no such kernel exists — as for $R [t]$ — expect the problem to be hard.

Move 3 makes the finiteness of $n$ explicit: the induction terminates after $n$ pivots. Nothing in the argument survives to matrices of infinite size, where the elimination never finishes.

Worked Example

A finite matrix ring

Take $R = ℤ / 12 ℤ$ , whose radical is $(2) \cap (3) = (6) = {0, 6}$ . Then

rad M_{2} (ℤ / 12 ℤ) = M_{2} ((6)),

(E.1)

Sixteen matrices, all four entries drawn from ${0, 6}$ .

Two checks. First, $M_{2} ((6))^{2} = M_{2} ((6) (6)) = M_{2} ((36)) = M_{2} (0) = 0$ , so the radical is nilpotent of index $2$ — as it must be, since $M_{2} (ℤ / 12)$ is finite hence artinian. Second, the unit test: $I_{2} + (\begin{matrix} 6 & 6 \\ 6 & 6 \end{matrix}) = (\begin{matrix} 7 & 6 \\ 6 & 7 \end{matrix})$ has determinant $49 - 36 = 13 \equiv 1 (\mod 12)$ , a unit, so the matrix is invertible.

The quotient is $M_{2} (ℤ / 12) / M_{2} ((6)) ≅ M_{2} (ℤ / 6 ℤ) ≅ M_{2} (𝔽_{2}) \times M_{2} (𝔽_{3})$ , which is semisimple — exactly what $(5 . M . 2) (1)$ predicts.

A local power series ring, then matrices over it

Let $R = ℤ_{(p)}$ , the localisation of $ℤ$ at the prime $p$ , a local ring with $rad R = p R$ and residue field $𝔽_{p}$ . Put $A = ℤ_{(p)} [[t]]$ . By the power series theorem,

rad A = p R + t A = p A + t A,

(E.2)

The ideal generated by $p$ and $t$ — every series whose constant term is divisible by $p$ .

Then $A / rad A ≅ R / p R ≅ 𝔽_{p}$ , so $A$ is local. Note that $rad A$ contains $t$ , which is not nilpotent, so $rad A$ is not nil and $A$ is not artinian — consistent with $(5 . M . 4)$ , and a reminder that local is far weaker than artinian local.

Now stack the constructions: $M_{3} (A)$ has $rad M_{3} (A) = M_{3} (p A + t A)$ and

M_{3} (A) / rad M_{3} (A) ≅ M_{3} (𝔽_{p}),

(E.3)

Simple artinian of $𝔽_{p}$ -dimension $9$ . So $M_{3} (A)$ is semilocal but not local and not artinian.

Process and Workflow

Peel off matrix layersReplace

M_{n} (S)

S

; the radical returns entrywise at the end. This shrinks the working dimension by a factor

n^{2}

Peel off complete or nilpotent layersReplace

S [[t]]

S

, or

S \to S / 𝔑

for a nilpotent ideal

𝔑

. Record each kernel; the final answer is their sum with the pullback of the base radical.

Reduce to a corner if possibleIf a full idempotent

e

is visible, work in

e R e

, which is often much smaller, and transport back by

(21.10)

Compute the base radicalNow the ring is small: a finite-dimensional algebra, a finite ring or a field. Use the standard algorithms.

Reassemble and verifyPush the answer back through the layers, then test

1 + rad \subseteq U

on a few elements.

Which route applies to your construction?

Matrices or a cornerUse the entrywise formula or

(21.10)

. The answer is exact, needs no hypotheses on the base ring, and preserves semiprimitivity in both directions.

Completion or nilpotent kernelUse

(4.5)

and

(4.6)

: the radical is the preimage of the base radical. Expect the result to be non-nil, so do not assume artinian behaviour.

Polynomial variables adjoinedOnly Amitsur's theorem is available: the radical is

N [T]

for a nil ideal

N

R

, and pinning down

N

is open in general.

Scalars extended along a field extensionSeparable algebraic extensions behave, by

(5.17)

; transcendental ones give only a nil intersection, by

(5.15)

. See the Radical under Field Extension page.

Comparison and Classification

The radical under standard constructions
Construction	Radical	Hypotheses	Reference
$M_{n} (R)$	$M_{n} (rad R)$	any $R$ , $n$ finite	§4, Example 7
$T_{n} (R)$ upper triangular	diagonal entries in $rad R$ , rest arbitrary	any $R$	(5.M.6)
$e R e$ , $e = e^{2}$	$e (rad R) e$	any $R$ , any idempotent	(21.10)
$R [[t]]$	$rad R + t R [[t]]$	any $R$	Ex. 5.6
$R [[x; σ]]$	$rad R + x R [[x; σ]]$	$σ \in Aut (R)$	(5.M.5)
$R \times S$	$rad R \times rad S$	any $R$ , $S$	componentwise units
$R [t]$	$N [t]$ with $N = R \cap rad R [t]$ nil	any $R$ ; $N$ not identified	(5.10)
$R [T]$ , $R$ commutative	$(Nil R) [T]$	$R$ commutative	(5.1)
$R \otimes_{k} K$	$(rad R) \otimes_{k} K$	$K / k$ separable algebraic	(5.17)

Which properties each construction preserves
	Radical in closed form	$R$ semiprimitive $\Rightarrow$ so is it	$rad R$ nilpotent $\Rightarrow$ its radical nilpotent	$R$ local $\Rightarrow$ so is it
$M_{n} (R)$ , $n \geq 2$	yes	yes	yes	no
$T_{n} (R)$ , $n \geq 2$	yes	no	yes	no
$e R e$ , $e \neq 0$	yes	yes	yes	yes, trivially
$R [[t]]$	yes	no	no	yes
$R [[x; σ]]$	yes	no	no	yes
$R [t]$	open	yes	yes	no

Which properties each construction preserves

The single row that matters most is the last: $R [t]$ preserves semiprimitivity (if $rad R = 0$ then $R$ has no nonzero nil ideal, so $rad R [t] = 0$ by Amitsur's theorem) yet its radical has no closed form. Preservation of a property and computability of the invariant are different questions. The nilpotence column reads yes for $R [t]$ for a cheap reason: $N = R \cap rad R [t]$ is nil, hence contained in $rad R$ by $(4.11)$ , so $rad R$ nilpotent forces $N [t]$ nilpotent. The $e R e$ entry in the last column is equally cheap: a local ring has no idempotents other than $0$ and $1$ , so $e R e = R$ .

Relationship Map

Both theorems on this page are instances of a single containment pattern, and the pattern is what to remember when meeting a new construction.

Computing $rad$ of a constructed ring — three routes, in order of preference
- Morita route — the construction is a matrix ring or a full corner
  - $rad M_{n} (R) = M_{n} (rad R)$
  - $rad (e R e) = e (rad R) e$
  - radical is a Morita invariant
- Filtration route — an ideal $𝔄$ with $1 + 𝔄 \subseteq U (A)$ and known quotient
  - $𝔄$ nilpotent: triangular rings, $R [t] / (t^{m})$
  - $𝔄$ topologically nilpotent: $R [[t]]$ , $R [[x; σ]]$
  - answer is $rad$ of the quotient, pulled back
- Neither route available — polynomial and group ring extensions
  - $rad R [t] = N [t]$ , $N$ nil — Amitsur
  - is $N = {Nil}^{*} R$ ? equivalent to Köthe's conjecture
  - $rad k G$ for infinite $G$ : hard, section 6

𝔄

ideal,

1 + 𝔄 \subseteq U (A)

𝔄 \subseteq rad A

(4.5)

rad A / 𝔄 = rad (A / 𝔄)

(4.6)

rad A

known

Applications and Industry Use

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

Symbolic computation

Dimension reduction before decomposition

A finite-dimensional algebra presented as $M_{n} (S)$ has $n^{2} \dim S$ structure constants. Since the radical is entrywise, a system computes $rad S$ and expands, replacing an $O ((n^{2} d)^{3})$ linear algebra problem by an $O (d^{3})$ one.

Modular representation theory

Coefficients in a complete local ring

Brauer theory works over a complete discrete valuation ring such as $ℤ_{p}$ or $ℤ_{(p)} [[t]]$ precisely because completion makes the coefficient ring local with a known radical, so reduction modulo the residue field is controlled.

Coding theory

Codes over chain and Galois rings

Linear codes over $ℤ / p^{m}$ and over finite chain rings are analysed via the filtration by powers of the radical; the matrix formula is what lets generator matrices over the ring be reduced to the residue field entry by entry.

Control and systems theory

Formal solutions and delay systems

Transfer function algebras over $k [[t]]$ and skew power series rings $k [[x; σ]]$ model time-varying and delay systems. Invertibility of a transfer matrix reduces, by the two theorems here, to invertibility of its constant term over the residue ring.

The honest summary is that these are structural results consumed inside algebra: they are what make radical a Morita invariant and what supply the standard examples of noncommutative local rings. Their engineering visibility is indirect, through computer algebra libraries and through coding and systems models built over local coefficient rings.

Standards and Notation

Standards here covers notation, symbol and markup standards, and reference implementations, rather than material or design codes.

Matrix ring

M_{n} (R)

in this collection;

R^{n \times n}

and

{Mat}_{n} (R)

occur elsewhere

Radical of it

M_{n} (rad R)

, equal to

(rad R) \cdot M_{n} (R)

Power series

R [[t]]

R [[x]]

; the double bracket is standard,

R [| t |]

is a rare variant

Twisted version

R [[x; σ]]

with

x r = σ (r) x

; some authors write

R [[x, σ]]

or place

σ

on the left

Constant term

f (0)

{ev}_{0} (f)

ε (f)

for the augmentation

ImplementationsPowerSeriesRing in Sage and Magma; MatrixAlgebra plus RadicalOfAlgebra in GAP

Computational Notes

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

**Never compute in $M_{n}$ directly.** Radical algorithms for a $d$ -dimensional algebra cost roughly $O (d^{3})$ field operations; applied to $M_{n} (S)$ with $\dim S = d$ that is $O (n^{6} d^{3})$ . Computing $rad S$ and expanding is a factor $n^{6}$ cheaper.
Power series are decidable at the constant term. Membership in $rad R [[t]]$ and invertibility in $R [[t]]$ both depend only on the coefficient of $t^{0}$ , so no truncation error is involved in either test.
Truncations agree. In $R [t] / (t^{m})$ the ideal $(t)$ is nilpotent, so the same argument gives $rad (R [t] / (t^{m})) = {f : f (0) \in rad R}$ . This is the representation a computer algebra system actually stores.
Library entry points. GAP exposes RadicalOfAlgebra, Magma JacobsonRadical, Sage A.radical(); all require a finite-dimensional algebra over a field, and all will happily accept a matrix algebra without exploiting the entrywise structure unless the base algebra is supplied separately.
No algorithm for the polynomial case. Since identifying $R \cap rad R [t]$ in general is equivalent to Köthe's conjecture, there is no procedure that returns generators of $rad R [t]$ for an arbitrary ring $R$ .

Failure Modes and Common Mistakes

Do not read the triangular formula as *all entries in $rad R$ *: the off-diagonal entries are unconstrained, and for $R$ a division ring the radical of $T_{n} (R)$ is the whole strictly upper triangular part, not zero.
Do not assume $R [[t]]$ inherits chain conditions from $R$ : for $R \neq 0$ it is never left artinian, because $t$ lies in the radical and is not nilpotent.
Do not apply $(4.6)$ to an ideal that is not inside the radical; the quotient rule fails badly otherwise, as $ℤ \to ℤ / 4$ shows.
Do not transplant the entrywise formula to rings of infinite matrices, or to non-unital rings of finitary matrices, without a fresh proof.

Quick Reference

Matrix ring

rad M_{n} (R) = M_{n} (rad R)

, any

R

n

finite

Matrix quotient

M_{n} (R) / rad ≅ M_{n} (R / rad R)

Power series

rad R [[t]] = rad R + t R [[t]]

Power series quotient

R [[t]] / rad ≅ R / rad R

Units

f \in U (R [[t]]) iff f (0) \in U (R)

Corner

rad (e R e) = e (rad R) e

Triangulardiagonal in

rad R

, off-diagonal free

Polynomial

rad R [t] = N [t]

N

nil — not identified

Property transfer at a glance
Question	$M_{n} (R)$	$R [[t]]$
Semiprimitive when $R$ is?	yes, and conversely	never, for $R \neq 0$
Local when $R$ is?	only for $n = 1$	yes, same residue ring
Left artinian when $R$ is?	yes	never, for $R \neq 0$
Radical nilpotent when $rad R$ is?	yes, same index	no, $t$ is not nilpotent
Radical meets $R$ in $rad R$ ?	yes, on scalar matrices	yes, on constants

Frequently Asked Questions

Why is the matrix formula entrywise while the power series formula constrains only the constant term?

Because the mechanisms differ. In $M_{n} (R)$ the matrix units conjugate any entry into the $(1, 1)$ position, so a two-sided ideal cannot contain a matrix without containing all matrices with those entries — membership is forced entry by entry. In $R [[t]]$ nothing moves coefficients around; instead the whole ideal $t R [[t]]$ is swallowed by the radical because $1 - t g$ is invertible by a convergent geometric series. The radical is then simply the preimage of $rad R$ .

Is the Jacobson radical a Morita invariant?

Yes. $rad M_{n} (R) = M_{n} (rad R)$ together with the corner ring theorem $rad (e R e) = e (rad R) e$ for a full idempotent $e$ gives invariance under the generators of Morita equivalence. Concretely, $R$ and $M_{n} (R)$ are semiprimitive together, semilocal together and semiperfect together, and their radical quotients correspond.

Does

R [[t]]

inherit chain conditions from

R

Not the descending one. For $R \neq 0$ the element $t$ lies in $rad R [[t]]$ and is not nilpotent, so the radical is not nilpotent and $R [[t]]$ cannot be left artinian, by $(4.12)$ — even when $R$ is a field. This is the standard reminder that a local ring need not be an artinian local ring.

What is the analogue for infinite matrix rings?

There is none of this form. Both proofs given here use $n < \infty$ : the row reduction terminates after $n$ pivots, and the ideal correspondence for $M_{n} (R)$ is a finite-matrix statement. Rings of row-finite or column-finite infinite matrices need separate analysis, and the naive entrywise guess should not be assumed.

Why does the polynomial ring behave so much worse than the power series ring?

Degrees add in $R [t]$ , so $1 - t g$ is essentially never a unit and no analogue of $t R [[t]]$ is available inside the radical. Amitsur's theorem $(5.10)$ still gives $rad R [t] = N [t]$ with $N = R \cap rad R [t]$ nil, but deciding whether $N$ is the upper nilradical ${Nil}^{*} R$ is Problem $(5.12)$ , equivalent to Köthe's conjecture.

Does the skew case need

σ

to be an automorphism?

The proof as given does. Bijectivity of $σ$ is what makes $x A = A x$ a two-sided ideal with quotient $R$ ; for a merely injective endomorphism the left and right ideals generated by $x$ can differ, and the clean preimage description is not justified without further hypotheses.

References

T. Y. Lam, A First Course in Noncommutative Rings, Graduate Texts in Mathematics 131, Springer-Verlag, 1991: §4, Example 7 following (4.15) (pp. 60–62) for the matrix ring; §5, Exercise 6 (p. 82) for the power series ring; §5 (pp. 70–81) for the surrounding change-of-rings results.
T. Y. Lam, A First Course in Noncommutative Rings, §21, (21.10)–(21.14), for the corner ring theorem and the idempotent-theoretic derivation of the matrix formula.
T. Y. Lam, Exercises in Classical Ring Theory, 2nd edition, Problem Books in Mathematics, Springer-Verlag, 2003, solutions to the exercises of §5.
N. Jacobson, Structure of Rings, American Mathematical Society Colloquium Publications 37, revised edition, 1964, Chapter I.
F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, 2nd edition, Graduate Texts in Mathematics 13, Springer-Verlag, 1992, §15 and §21–22 for the radical and Morita equivalence.
S. A. Amitsur, “Radicals of polynomial rings”, Canadian Journal of Mathematics 8 (1956), 355–361.

AI Suggested Questions

Prove the corner ring theorem $rad (e R e) = e (rad R) e$ and deduce the matrix formula from it.
What is the Jacobson radical of the ring of row-finite $ω \times ω$ matrices over a division ring?
Give a ring $R$ with $rad R \neq 0$ but $rad R [t] = 0$ , and explain what Amitsur's theorem says about it.
Compute $rad R [[x; σ]]$ when $σ$ has finite order, and compare with the fixed subring $R^{σ}$ .
Show that $R$ and $M_{n} (R)$ are semiperfect together, and identify what happens to idempotent lifting.
Why is Problem $(5.12)$ equivalent to Köthe's conjecture, and what would a counterexample look like?
Determine the radical of the Laurent series ring $R ((t))$ when $R$ is a division ring, and when $R$ is local.
How do these formulas interact with completion: is $rad$ of an inverse limit the inverse limit of the radicals?

Radical of Matrix and Power Series Rings

Executive Summary

Overview

Learning Objectives

Definitions

Core Concepts

Two mechanisms for transporting the radical

Matrix units move entries

Geometric series converge

Why the second mechanism gives a preimage

Key Results

Proof Techniques and Method

Squeeze with $(4.5)$ then $(4.6)$

Conjugate by matrix units

Row reduce inside the radical

Worked Example

A finite matrix ring

A local power series ring, then matrices over it

Process and Workflow

Comparison and Classification

Relationship Map

Applications and Industry Use

Dimension reduction before decomposition

Coefficients in a complete local ring

Codes over chain and Galois rings

Formal solutions and delay systems

Standards and Notation

Computational Notes

Failure Modes and Common Mistakes

Quick Reference

Frequently Asked Questions

References

AI Suggested Questions

Continue learning

01Executive Summary

02Overview

03Learning Objectives

04Definitions

05Core Concepts

Two mechanisms for transporting the radical

Matrix units move entries

Geometric series converge

Why the second mechanism gives a preimage

06Key Results

07Proof Techniques and Method

Squeeze with (4.5) then (4.6)

Conjugate by matrix units

Row reduce inside the radical

08Worked Example

A finite matrix ring

A local power series ring, then matrices over it

09Process and Workflow

10Comparison and Classification

11Relationship Map

12Applications and Industry Use

Dimension reduction before decomposition

Coefficients in a complete local ring

Codes over chain and Galois rings

Formal solutions and delay systems

13Standards and Notation

14Computational Notes

15Failure Modes and Common Mistakes

16Quick Reference

17Frequently Asked Questions

18Related KEVOS Topics

19References

20AI Suggested Questions

Continue learning

Executive Summary

Overview

Learning Objectives

Definitions

Core Concepts

Key Results

Proof Techniques and Method

Squeeze with $(4.5)$ then $(4.6)$

Worked Example

Process and Workflow

Comparison and Classification

Relationship Map

Applications and Industry Use

Standards and Notation

Computational Notes

Failure Modes and Common Mistakes

Quick Reference

Frequently Asked Questions

Related KEVOS Topics

References

AI Suggested Questions