Projective Modules over Local Rings Are Free

Q: Why is the rank well defined?

Because R/rad R is a division ring, and modules over a division ring have a unique dimension. This is exactly the step that fails for a general ring, and it is why local rings have invariant basis number while some other rings do not.

Q: What is the analogue for a ring that is not local?

For a semiperfect ring, R/rad R is semisimple, so a finitely generated projective is determined by the multiplicities of the simple summands of its reduction. Translated back, every finitely generated projective is a direct sum of principal indecomposable modules, uniquely up to permutation.

Q: Can I replace projective by flat?

No. Q is flat over the local ring Z_(p) and not free. Projectivity is used twice in the proof — to construct the lift and to split the surjection — and flatness supplies neither.

Q: How does this give Dickson's theorem?

Restrict a principal indecomposable kG-module to a Sylow p-subgroup H. Since kG is free over kH, the restriction is finitely generated projective, and kH is local because H is a p-group in characteristic p. So the restriction is free over kH, and comparing dimensions gives |H| _k U.

Executive Summary

Over a general ring, projective and free are far apart: a nonprincipal ideal of a Dedekind domain and the column module of $M_{n} (k)$ are both projective and neither is free. Over a local ring the gap closes completely in the finitely generated range: every finitely generated projective right module is free $(19.29)$ .

The mechanism is simple and worth internalising. Reduction modulo $J = rad R$ turns a projective module into a vector space over the division ring $R / J$ , where the only invariant is dimension. A lifting lemma $(19.27)$ says that reduction does not lose information on finitely generated projectives: $P ≅ Q$ if and only if $P / P J ≅ Q / Q J$ . Comparing $P$ with $R^{n}$ for $n = \dim_{R / J} P / P J$ finishes the argument.

$P ≅ R^{n}$ Conclusion

$\dim_{R / J} P / P J$ The rank

2Uses of Nakayama

1958Kaplansky drops f.g.

Overview

Projective modules are the objects that make homological algebra work, and the practical question about any ring is how far its projectives are from free. That distance is measured by $K_{0} (R)$ modulo the class of $R$ ; for a local ring the distance is zero.

(R, J) local, P_{R} f.g. projective ⟹ P ≅ R^{n}, n = \dim_{R / J} (P / P J) .

(19.29)

The rank is well defined because $R / J$ is a division ring, so its modules have a unique dimension.

The lemma is stated for any ideal $J \subseteq rad R$ , not only for the radical itself, and the extra generality is used later: over a semiperfect ring the same lemma reduces the classification of projectives to the semisimple quotient, where Wedderburn–Artin takes over and the answer is a direct sum of principal indecomposables rather than a free module.

Learning Objectives

State $(19.27)$ with its hypotheses: $J \subseteq rad R$ , both modules finitely generated and projective.
Reproduce the proof of $(19.27)$ , naming the two Nakayama steps.
Prove $(19.29)$ and compute the rank of $P$ from $P / P J$ .
Explain why finite generation cannot simply be dropped from the lemma, and what Kaplansky proved instead.
Deduce invariant basis number for local rings and compute $K_{0}$ .
Prove Dickson's theorem on the dimensions of principal indecomposable $k G$ -modules.

Definitions

Projective: $P_{R}$ is projective if it is a direct summand of a free module; equivalently every surjection $M ↠ P$ splits; equivalently ${Hom}_{R} (P, -)$ is exact.
$P J$: The submodule of $P$ generated by all $x j$ with $x \in P$ , $j \in J$ . For $P$ projective, $P / P J$ is a projective $R / J$ -module.
Nakayama's Lemma: If $M_{R}$ is finitely generated and $M rad R = M$ , then $M = 0$ . Equivalently $M rad R ⊊ M$ for $M \neq 0$ finitely generated.
Free of rank $n$: Isomorphic to $R^{n}$ . Over a ring with invariant basis number, $n$ is an invariant of the module.
$K_{0} (R)$: The Grothendieck group of finitely generated projective modules under direct sum.
Principal indecomposable: An indecomposable direct summand of $R_{R}$ over a one-sided artinian ring; the building block of projectives when $R$ is not local.

Modules are right modules. Everything on this page holds verbatim for left modules, because locality is left-right symmetric.

Core Concepts

Why reduction modulo the radical is the right move

Over the division ring $R / J$ every module is free and dimension is a complete invariant. So if reduction modulo $J$ can be reversed on the class of objects we care about, classification over $R$ collapses to counting a dimension. That reversal is exactly what $(19.27)$ provides, and it works for one reason only: projectivity supplies the lift, and Nakayama supplies the injectivity and surjectivity of the lift.

P

f.g. projective

P / P J

over

R / J

dimension

n

P ≅ R^{n}

The two Nakayama steps

The first use makes the lift surjective: if $im (f) + Q J = Q$ then $(Q / im f) J = Q / im f$ , and $Q$ finitely generated forces $im (f) = Q$ . The second use makes it injective: the kernel splits off as a direct summand $P'$ , reduction shows $P' / P' J = 0$ , and $P'$ finitely generated forces $P' = 0$ . Both steps need finite generation, and both fail without it.

What survives when the ring is not local

If $R$ is only semiperfect, $R / rad R$ is semisimple rather than a division ring, so $P / P rad R$ is a direct sum of simple modules with multiplicities instead of a single dimension. The same lemma then says that a finitely generated projective is determined by that multiset of multiplicities — which is precisely the statement that projectives are direct sums of principal indecomposables, uniquely.

Key Results

Lemma(19.27)Reduction is faithful on finitely generated projectives

Let $R$ be a ring, let $J$ be an ideal of $R$ with $J \subseteq rad R$ , and write $\bar{R} = R / J$ . Let $P, Q$ be finitely generated projective right $R$ -modules. Then $P ≅ Q$ as $R$ -modules if and only if $P / P J ≅ Q / Q J$ as $\bar{R}$ -modules.

Proof

The only if direction is immediate: an isomorphism $P \to Q$ carries $P J$ onto $Q J$ and so induces an isomorphism of the quotients.

For the converse, let $\bar{f} : P / P J \to Q / Q J$ be an $\bar{R}$ -isomorphism. Composing the projection $P ↠ P / P J$ with $\bar{f}$ gives a map $P \to Q / Q J$ ; since $P$ is projective and $Q ↠ Q / Q J$ is onto, this lifts to an $R$ -homomorphism $f : P \to Q$ inducing $\bar{f}$ .

Surjectivity (first Nakayama step). Because $\bar{f}$ is onto, $im (f) + Q J = Q$ . Hence the finitely generated module $Q / im (f)$ satisfies $(Q / im f) J = Q / im f$ , and $J \subseteq rad R$ , so Nakayama's Lemma $(4.22)$ gives $Q / im (f) = 0$ , i.e. $f$ is onto.

Injectivity (second Nakayama step). As $Q$ is projective, the surjection $f$ splits: $P = P' \oplus Q'$ with $P' = \ker (f)$ and $f |_{Q'} : Q' \to Q$ an isomorphism. Reducing modulo $J$ gives $P / P J ≅ P' / P' J \oplus Q' / Q' J$ , and under this decomposition $\bar{f}$ kills $P' / P' J$ and maps $Q' / Q' J$ isomorphically onto $Q / Q J$ . Since $\bar{f}$ is injective, $P' / P' J = 0$ , i.e. $P' = P' J$ . But $P'$ is a direct summand of the finitely generated module $P$ , hence finitely generated, so Nakayama gives $P' = 0$ . Therefore $\ker (f) = 0$ and $f$ is an isomorphism.

Theorem(19.29)Finitely generated projectives over a local ring are free

Let $(R, J)$ be a local ring, $J = rad R$ . Then every finitely generated projective right $R$ -module $P$ is free, of rank $n = \dim_{R / J} (P / P J)$ .

Proof

By $(19.1)$ the ring $\bar{R} = R / J$ is a division ring, so the finitely generated $\bar{R}$ -module $P / P J$ is free of some finite rank $n$ : $P / P J ≅ {\bar{R}}^{n}$ .

Put $Q = R^{n}$ , a finitely generated projective right $R$ -module with $Q / Q J ≅ (R / J)^{n} = {\bar{R}}^{n} ≅ P / P J$ . Both $P$ and $Q$ are finitely generated projective and $J \subseteq rad R$ , so $(19.27)$ applies and gives $P ≅ Q = R^{n}$ .

Corollary—Invariant basis number and

K_{0}

A local ring $R$ has invariant basis number: if $R^{n} ≅ R^{m}$ then reducing modulo $J$ gives ${\bar{R}}^{n} ≅ {\bar{R}}^{m}$ as vector spaces over the division ring $\bar{R}$ , so $n = m$ . Consequently every finitely generated projective has a well-defined rank, and $K_{0} (R) ≅ ℤ$ , generated by the class of $R$ .

Remark—Kaplansky's extension

Kaplansky proved in 1958 that over a local ring every projective module is free, with no finiteness hypothesis. That result is genuinely harder: the proof above uses Nakayama twice, and Nakayama requires finite generation. Kaplansky's argument instead decomposes an arbitrary projective into a direct sum of countably generated modules and then handles the countably generated case directly.

Theorem(19.30)Dickson's divisibility theorem

Let $k$ be a field of characteristic $p > 0$ , let $G$ be a finite group, and put $R = k G$ . Let $H \leq G$ be a Sylow $p$ -subgroup. Then for every principal indecomposable right $R$ -module $U$ — that is, every indecomposable direct summand of $R_{R}$ — the order $| H |$ divides $\dim_{k} U$ .

Proof

$R = k G$ is a finite-dimensional algebra, hence right artinian, so $R_{R}$ has finite length and $(19.22)$ gives a Krull–Schmidt decomposition $R_{R} = U_{1} \oplus \dots \oplus U_{n}$ into principal indecomposables. Each $U_{i}$ is a direct summand of $R_{R}$ and therefore projective.

Choose left coset representatives $g_{1}, \dots, g_{m}$ for $H$ in $G$ , $m = [G : H]$ . Then $k G = g_{1} k H \oplus \dots \oplus g_{m} k H$ as right $k H$ -modules, so $R$ restricted to $k H$ is free of rank $m$ . Each $U_{i}$ is a direct summand of that free $k H$ -module, hence a finitely generated projective right $k H$ -module.

Since $H$ is a $p$ -group and $char k = p$ , the group algebra $k H$ is a local ring $(19.10)$ . By $(19.29)$ each $U_{i}$ is therefore free over $k H$ , say of rank $r_{i}$ . Comparing $k$ -dimensions, $\dim_{k} U_{i} = r_{i} \cdot | H |$ , so $| H | ∣ \dim_{k} U_{i}$ .

Corollary—Projectives over artinian rings

Over a right artinian ring $R$ , every finitely generated projective right module is isomorphic to a finite direct sum of principal indecomposable right $R$ -modules, and by $(19.23)$ the multiplicities are uniquely determined. Freeness is the special case in which there is only one principal indecomposable — which by $(19.19)$ happens exactly when $R$ is local.

Proof Techniques and Method

How these proofs work, and which move to reuse.

Move 1

Lift along the radical

Projectivity converts a map defined on $P / P J$ into a map defined on $P$ . This is the only role projectivity plays, and it is why the lemma is about projectives rather than arbitrary modules.

Move 2

Nakayama upgrades mod- $J$ facts

*Onto modulo $J$ * becomes onto, and *zero modulo $J$ * becomes zero, provided the module in question is finitely generated. Nearly every semiperfect-ring argument is this move repeated.

Move 3

Compare against a chosen model

To identify an unknown $P$ , pick a model $Q$ with the same reduction and quote faithfulness. Over a local ring the models are $R^{n}$ ; over a semiperfect ring they are sums of principal indecomposables.

There is a second, entirely different route to $(19.29)$ : apply the Krull–Schmidt–Azumaya theorem. A finitely generated projective $P$ is a direct summand of some $R^{n}$ , and $R$ is strongly indecomposable as a module over the local ring $R$ , so uniqueness of decompositions forces $P$ to be a sum of copies of $R$ . That proof trades Nakayama for Azumaya and is worth knowing because it generalises differently.

Worked Example

Principal indecomposables of $𝔽_{3} S_{3}$

Take $k = 𝔽_{3}$ , $G = S_{3}$ , $R = k G$ , so $\dim_{k} R = 6$ and $p = 3$ . The Sylow $3$ -subgroup is $H = A_{3} ≅ C_{3}$ , which is normal, of order $3$ and index $2$ .

First, $k H ≅ k [t] / (t^{3})$ via $t = h - 1$ , since $(h - 1)^{3} = h^{3} - 1 = 0$ in characteristic $3$ . This is a local ring with radical $(t)$ and residue field $k$ — the case of $(19.10)$ for a cyclic $p$ -group.

Because $H$ is a normal $p$ -subgroup and $| G / H | = 2$ is invertible in $k$ , the ideal generated by ${h - 1 : h \in H}$ is nilpotent with semisimple quotient, so

R / rad R ≅ k [G / H] = 𝔽_{3} C_{2} ≅ 𝔽_{3} [x] / (x^{2} - 1) ≅ 𝔽_{3} \times 𝔽_{3},

(E.1)

$\dim_{k} rad R = 6 - 2 = 4$ . There are exactly two simple $R$ -modules, the trivial and the sign module, each of $k$ -dimension $1$ .

Idempotents lift modulo the nilpotent ideal $rad R$ , so the splitting $𝔽_{3} \times 𝔽_{3}$ lifts to $1 = e_{1} + e_{2}$ with $e_{1}, e_{2}$ orthogonal primitive idempotents, and $R_{R} = U_{1} \oplus U_{2}$ with $U_{i} = e_{i} R$ . Restricting to $k H$ , $R$ is free of rank $[G : H] = 2$ , so each $U_{i}$ is finitely generated projective over the local ring $k H$ , hence free by $(19.29)$ .

6 = \dim_{k} U_{1} + \dim_{k} U_{2}, 3 ∣ \dim_{k} U_{i} and \dim_{k} U_{i} > 0 ⟹ \dim_{k} U_{1} = \dim_{k} U_{2} = 3 .

(E.2)

So each principal indecomposable is free of rank $1$ over $k H$ , of $k$ -dimension $3$ — exactly the divisibility Dickson's theorem predicts. As a cross-check, $2 = r_{1} + r_{2}$ matches the rank of $R$ over $k H$ .

A contrast: $M_{2} (k)$

Let $S = k^{2}$ be the module of columns over $R = M_{2} (k)$ . Then $R_{R} ≅ S \oplus S$ , so $S$ is projective with $\dim_{k} S = 2$ , while every free module has $k$ -dimension a multiple of $4$ . So $S$ is projective and not free. Consistently, $M_{2} (k)$ is not local: $E_{11}$ is a nontrivial idempotent.

Process and Workflow

Confirm the ring is localCheck

R / rad R

is a division ring, or for an artinian ring check that the only idempotents are

0

and

1

(19.19)

Reduce the moduleForm

P / P rad R

and compute its dimension over the residue division ring. This is a linear-algebra step.

Read off the rankThat dimension is the free rank

n

; it is well defined because the residue ring is a division ring.

Lift the identificationApply

(19.27)

with

Q = R^{n}

to conclude

P ≅ R^{n}

over

R

, not merely modulo the radical.

My finitely generated projective is not free. What went wrong?

$R$ is not localExpected. Decompose

R / rad R

into simple factors; each factor supplies a principal indecomposable and freeness is only the one-factor case.

$R$ is local but the module is not finitely generatedIt is still free, but by Kaplansky's theorem rather than by

(19.29)

— the Nakayama argument does not apply.

The module is flat, not projectiveFlat does not imply free over a local ring:

ℚ

is a flat, non-free

ℤ_{(p)}

-module.

Comparison and Classification

Are finitely generated projectives free?
Ring	f.g. projective $\Rightarrow$ free?	Reason
Local ring $(R, J)$	yes	(19.29); all projectives, by Kaplansky
Division ring $D$	yes	every module is free
$k [[x]]$ , $ℤ_{(p)}$ , $ℤ_{p}$	yes	local
$k G$ , $G$ a finite $p$ -group, $char k = p$	yes	local by (19.10)
Principal ideal domain	yes	f.g. torsion-free is free
$k [x_{1}, \dots, x_{n}]$	yes	Quillen–Suslin theorem
$ℤ [\sqrt{- 5}]$	no	nonprincipal ideals are projective
$M_{n} (k)$ , $n \geq 2$	no	the column module is projective, not free
$ℤ / 6 ℤ$	no	$ℤ / 2 ℤ$ is a projective summand
General semiperfect ring	no	sums of principal indecomposables instead

Which hypotheses each conclusion needs
	$J \subseteq rad R$	$P, Q$ projective	$P, Q$ f.g.	$R / J$ a division ring
(19.27) faithfulness	yes	yes	yes	no
(19.29) freeness	yes	yes	yes	yes
Kaplansky's theorem	yes	yes	no	yes
Sums of principal indecomposables	yes	yes	yes	no

Which hypotheses each conclusion needs

Relationship Map

The result sits at the junction of three streams: Nakayama-style radical arguments, Krull–Schmidt uniqueness, and the classification of projectives that becomes $K$ -theory.

(19.29) f.g. projective is free — over a local ring $(R, rad R)$
- depends on
  - Nakayama's Lemma (4.22)
  - the lifting lemma (19.27)
  - $R / rad R$ a division ring (19.1)
- implies
  - local rings have invariant basis number
  - $K_{0} (R) ≅ ℤ$ for $R$ local
  - Dickson's theorem (19.30)
  - the trivial module is non-projective when $p ∤ 1$
- generalises to
  - Kaplansky: all projectives are free
  - semiperfect rings: sums of principal indecomposables
  - projective covers and §24 of the second course

R

local

R / rad R

division ringreduction faithful on f.g. projectivesprojectives free

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

Dimensions of projectives

Dickson's theorem $(19.30)$ constrains the dimensions of principal indecomposable $k G$ -modules by the order of a Sylow $p$ -subgroup, a constraint used constantly when computing decomposition and Cartan matrices.

Algebraic K-theory

The base case of $K_{0}$

$K_{0}$ of a local ring is $ℤ$ . Local triviality is what makes $K_{0}$ of a scheme a sheaf-theoretic invariant: projective modules are locally free, and the global obstruction is what the theory measures.

Commutative algebra and geometry

Locally free means projective

The statement that finitely generated projectives over a commutative ring are exactly the locally free modules of finite rank rests on this theorem applied at each localisation.

Symbolic computation

Deciding freeness

Over a finite-dimensional local algebra, checking that a projective is free reduces to a dimension count modulo the radical; over non-local algebras the same computation returns the multiplicities of principal indecomposables instead.

The honest framing is that this theorem is the local model in a local-to-global strategy. It is invoked not for its own sake but to say that all difficulty in classifying projectives is global.

Design Considerations

Design considerations here means the choices made when modelling a problem with these algebraic structures.

Which ideal to reduce by. $(19.27)$ allows any $J \subseteq rad R$ . Taking $J = rad R$ gives the strongest quotient; taking a smaller $J$ — for instance a power of the radical — is useful when you want to keep some nilpotent information.
Finite generation is a modelling choice, not a technicality. If your projectives are genuinely infinitely generated, cite Kaplansky and expect a different proof, not a longer one.
Which side. Locality is symmetric, so the choice of right modules is free; but if you later pass to a semiperfect or perfect ring, sidedness starts to matter and the choice should be fixed early.
Local versus semiperfect. If your ring has several principal indecomposables, do not force freeness. The correct target statement is a direct sum of principal indecomposables with well-defined multiplicities.

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, deciding whether a finitely generated projective is free costs a radical computation followed by a dimension count in $A / rad A$ ; no module-level isomorphism test is required, which is the practical content of $(19.27)$ .
Constructing an explicit basis, rather than proving one exists, means lifting a basis of $P / P J$ and verifying independence — the constructive shadow of the two Nakayama steps.
GAP's and Magma's decomposition routines for modules over a finite-dimensional algebra return projective indecomposables with multiplicities; over a local algebra that list has one entry and the multiplicity is the rank.
Freeness of projectives over polynomial rings — the Quillen–Suslin theorem — is a genuinely harder computational problem, with algorithms that produce a basis but at substantially higher cost than the local case.

Failure Modes and Common Mistakes

Do not conclude $P ≅ Q$ from $P / P J ≅ Q / Q J$ without checking that both are projective; the lemma is false for arbitrary finitely generated modules, as $ℤ / p ℤ$ and $ℤ / p^{2} ℤ$ show over $ℤ_{(p)}$ .
Do not assume invariant basis number for a general ring; it holds here because $R / J$ is a division ring, and it genuinely fails for some rings.
Do not read Dickson's theorem as a statement about all indecomposable $k G$ -modules; it is about principal indecomposables, i.e. the projective ones.
Do not use $(19.29)$ to claim freeness of a projective over a completely primary ring's quotient without rechecking that the quotient is still local.

Best Practices

Quote $(19.27)$ rather than $(19.29)$ whenever the ring is semiperfect but not local; the lemma is the general statement and the theorem is a special case.
State the rank as $\dim_{R / J} (P / P J)$ so that its well-definedness is visible.
When applying the theorem to a subalgebra — as in the Sylow restriction in Dickson's proof — verify that restriction really does turn the module into a finitely generated projective over the smaller ring.
Record whether you need Kaplansky's version; if your module is finitely generated, do not invoke the stronger theorem unnecessarily.

Quick Reference

Lemma (19.27)

P ≅ Q \Leftrightarrow P / P J ≅ Q / Q J

, for f.g. projectives,

J \subseteq rad R

Theorem (19.29)

(R, J)

local,

P

f.g. projective

\Rightarrow P ≅ R^{n}

Rank

n = \dim_{R / J} (P / P J)

IBNLocal rings have invariant basis number

$K_{0}$

K_{0} (R) ≅ ℤ

for

R

local

Kaplansky 1958Drop finite generation: all projectives are free

Dickson (19.30)

| H | ∣ \dim_{k} U

for principal indecomposable

U

Non-local analogueSums of principal indecomposables, unique multiplicities

Numbered results used on this page
Reference	Statement	Hypotheses
(4.22)	$M J = M \Rightarrow M = 0$	$M$ f.g., $J \subseteq rad R$
(19.27)	Reduction mod $J$ is faithful	$P, Q$ f.g. projective, $J \subseteq rad R$
(19.29)	f.g. projective is free	$(R, J)$ local
(19.10)	$k G$ is local	$char k = p$ , $G$ a finite $p$ -group
(19.30)	$\| H \|$ divides $\dim_{k} U$	$U$ principal indecomposable, $H$ Sylow $p$
(19.23)	Multiplicities are unique	$M$ f.g., $R$ right artinian

Frequently Asked Questions

Does the theorem need the module to be finitely generated?

The proof does, in two places, because Nakayama's Lemma does. The statement does not: Kaplansky proved in 1958 that every projective module over a local ring is free. His argument reduces to countably generated modules and does not go through Nakayama.

Why is the rank well defined?

Because $R / rad R$ is a division ring, and modules over a division ring have a unique dimension. This is exactly the step that fails for a general ring, and it is why local rings have invariant basis number while some other rings do not.

What is the analogue for a ring that is not local?

For a semiperfect ring, $R / rad R$ is semisimple, so a finitely generated projective is determined by the multiplicities of the simple summands of its reduction. Translated back, every finitely generated projective is a direct sum of principal indecomposable modules, uniquely up to permutation.

Can I replace projective by flat?

No. $ℚ$ is flat over the local ring $ℤ_{(p)}$ and not free. Projectivity is used twice in the proof — to construct the lift and to split the surjection — and flatness supplies neither.

How does this give Dickson's theorem?

Restrict a principal indecomposable $k G$ -module to a Sylow $p$ -subgroup $H$ . Since $k G$ is free over $k H$ , the restriction is finitely generated projective, and $k H$ is local because $H$ is a $p$ -group in characteristic $p$ . So the restriction is free over $k H$ , and comparing dimensions gives $| H | ∣ \dim_{k} U$ .

Is there a proof that avoids Nakayama?

Yes. A finitely generated projective is a direct summand of $R^{n}$ ; over a local ring $R$ is strongly indecomposable as a module over itself, so the Krull–Schmidt–Azumaya theorem forces every summand of $R^{n}$ to be a direct sum of copies of $R$ . This is Exercise 19.11 in Lam.

References

T. Y. Lam, A First Course in Noncommutative Rings, Graduate Texts in Mathematics 131, Springer-Verlag, 1991, §19 (pp. 294–310), especially (19.27)–(19.30).
I. Kaplansky, “Projective modules”, Annals of Mathematics 68 (1958), 372–377.
F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, 2nd edition, Graduate Texts in Mathematics 13, Springer-Verlag, 1992 (projective covers and semiperfect rings).
T. Y. Lam, Lectures on Modules and Rings, Graduate Texts in Mathematics 189, Springer-Verlag, 1999 (projective modules and invariant basis number).
C. W. Curtis and I. Reiner, Methods of Representation Theory, Volume I, Wiley, 1981 (principal indecomposable modules and modular representations).

AI Suggested Questions

Work through Kaplansky's 1958 proof that arbitrary projective modules over a local ring are free.
How does $(19.27)$ generalise to semiperfect rings, and what replaces the free rank?
Give an explicit example of a ring without invariant basis number and explain why it cannot be local.
Compute the principal indecomposable modules and their dimensions for $k G$ with $G = A_{4}$ and $char k = 2$ .
What is the relationship between projective covers and the lifting lemma used here?
Why is the Quillen–Suslin theorem so much harder than the local case, given that polynomial rings are localised into local rings?
For which finite-dimensional algebras does every finitely generated projective module happen to be free without the algebra being local?

01Executive Summary

02Overview

03Learning Objectives

04Definitions

05Core Concepts

Why reduction modulo the radical is the right move

The two Nakayama steps

What survives when the ring is not local

06Key Results

07Proof Techniques and Method

Lift along the radical

Nakayama upgrades mod-J facts

Compare against a chosen model

08Worked Example

Principal indecomposables of 𝔽3S3

A contrast: M2(k)

09Process and Workflow

10Comparison and Classification

11Relationship Map

12Applications and Industry Use

Dimensions of projectives

The base case of K0

Locally free means projective

Deciding freeness

13Design Considerations

14Computational Notes

15Failure Modes and Common Mistakes

16Best Practices

17Quick Reference

18Frequently Asked Questions

19Related KEVOS Topics

20References

21AI Suggested Questions

Continue learning

Executive Summary

Overview

Learning Objectives

Definitions

Core Concepts

Key Results

Proof Techniques and Method

Nakayama upgrades mod- $J$ facts

Worked Example

Principal indecomposables of $𝔽_{3} S_{3}$

A contrast: $M_{2} (k)$

Process and Workflow

Comparison and Classification

Relationship Map

Applications and Industry Use

The base case of $K_{0}$

Design Considerations

Computational Notes

Failure Modes and Common Mistakes

Best Practices

Quick Reference

Frequently Asked Questions

Related KEVOS Topics

References

AI Suggested Questions