Left Primitive Rings That Are Not Right Primitive

Q: Is there a concrete description of a left primitive ring that is not right primitive?

Bergman's construction from the mid-1960s is the original, and Jategaonkar produced others afterwards. Neither is reproduced in Lam's §11 — the text states their existence and points to the literature. What Lam does supply are the structural constraints such a ring must satisfy and the machinery used to build socle-free primitive rings.

Q: Why must a one-sided example have zero socle?

Because a left primitive ring is prime, and a prime ring with a minimal left ideal is both left and right primitive by (11.11) — the same conclusion follows from a minimal right ideal. So the presence of any minimal one-sided ideal forces symmetry, and both socles must vanish for asymmetry to be possible.

Q: If the free algebra is left primitive, why is it not a counterexample to symmetry?

Because reversing words is an anti-automorphism of k x_i, so the ring is isomorphic to its opposite and the left-handed result transports to the right. The theorems of Samuel and Formanek are constructions of primitive rings, not of asymmetric ones.

Q: Does the asymmetry affect the Jacobson radical?

No. The left primitive ideals and the right primitive ideals are genuinely different families, but both intersect to rad R. This is the clearest illustration that a symmetric invariant can be assembled from asymmetric ingredients.

Q: What is the role of the countability hypothesis in Formanek's theorem?

The construction enumerates the nonzero elements of the ring by the nonidentity elements of one free factor, so the coefficient ring must not be too large: the hypothesis in (11.29) is |k| ≤ |G|, which for a free ring on at least two generators is implied by countability of k. Later work removed the restriction for free algebras over arbitrary fields.

Q: Can a left primitive ring have several non-isomorphic faithful simple left modules?

Yes, and this is common in the socle-free world. Lam's examples include differential and skew polynomial rings over division rings, where the isomorphism classes correspond to conjugacy classes of parameters, and free algebras, for which two explicit non-isomorphic faithful simple modules are constructed in the exercises.

Executive Summary

A ring is left primitive when it has a faithful simple left module. Nothing in that definition refers to right modules, and the resulting notion is genuinely one-sided: G. Bergman constructed in the mid-1960s a ring primitive on one side only, and A. V. Jategaonkar produced further examples afterwards.

Such rings are severely constrained. A left primitive ring that is not right primitive must be prime, semiprimitive, noncommutative, artinian on neither side, and — the sharpest restriction — it must have zero socle on both sides, because a prime ring with a minimal one-sided ideal is primitive on both sides. Everything with a minimal one-sided ideal is symmetric; asymmetry lives only in the socle-free world.

Not symmetricPrimitivity

Symmetric

rad R

$0$ Socle of any example

1964Bergman

Overview

Most invariants in this subject are defined by one-sided data and turn out to be side-neutral anyway. The Jacobson radical is the model case: an intersection of maximal left ideals which coincides with the intersection of maximal right ideals. Primitivity looks similar in form and behaves completely differently.

rad R = ⋂ {left primitive ideals} = ⋂ {right primitive ideals}

(11.5)

The two families of ideals are different, yet their intersections agree — asymmetry that vanishes on intersection.

The practical questions are therefore: what tools produce left primitivity without any right-handed shadow, and where in the landscape of rings can a one-sided example possibly sit? Lam answers the first with the comaximality criterion $(11.28)$ and the free-product machinery of Formanek $(11.29)$ ; the second is answered by the results on minimal ideals and artinian rings.

Free algebras are the standard supply of socle-free primitive rings, and $(11.23)$ – $(11.27)$ prove them left primitive. They are not themselves counterexamples to symmetry — word reversal makes a free algebra isomorphic to its own opposite — but they are the environment in which counterexamples are built. The catalogue of such objects is collected in Catalogue of Counterexamples; the wider question of which properties transfer between sides is Left–Right Symmetry: What Transfers and What Does Not.

Learning Objectives

State precisely what left and right primitivity assert, and how each relates to $R^{op}$ .
Prove the criterion $(11.28)$ : left primitivity is the existence of a left ideal comaximal with every nonzero ideal.
Prove that a one-sided primitive ring is socle-free on both sides.
Follow Samuel's shift construction $(11.23)$ showing $k ⟨ x, y ⟩$ is left primitive.
State Formanek's free-ring theorem $(11.27)$ and semigroup-ring theorem $(11.29)$ with full hypotheses.
Classify common ring properties as side-symmetric or not.

Definitions

Definition(11.2)Left and right primitive

A ring $R$ is left primitive if it has a faithful simple left $R$ -module, and right primitive if it has a faithful simple right $R$ -module. Equivalently, $R$ is right primitive precisely when $R^{op}$ is left primitive. An ideal $𝔭 \subseteq R$ is left primitive if $R / 𝔭$ is a left primitive ring, equivalently if $𝔭 = ann (M)$ for some simple left $R$ -module $M$ .

Comaximal: $𝔄 + 𝔅 = R$ . In $(11.28)$ , $𝔄$ is a left ideal and $𝔅$ ranges over nonzero two-sided ideals.
$soc (_{R} R)$: The left socle: the sum of all minimal left ideals, or $0$ if there are none. The right socle is defined symmetrically; the two coincide for semiprime rings.
Free product $A * B$: For semigroups with identity $A, B \neq {1}$ , the semigroup of reduced words whose letters alternate between $A^{*} = A ∖ {1}$ and $B^{*} = B ∖ {1}$ .
Word type: A reduced word has type $A B$ if it starts with a letter of $A^{*}$ and ends with a letter of $B^{*}$ ; types $A A$ , $B A$ , $B B$ are defined the same way.
$max-supp (r)$: For $0 \neq r$ in a semigroup ring, the elements of maximal length occurring in the support of $r$ .

Semigroups here are understood to contain an identity element, and the free product is taken in that category.

Core Concepts

Why the definition can be asymmetric at all

Passing to $R^{op}$ turns right modules into left modules, so *right primitive for $R$ * is *left primitive for $R^{op}$ *. That is a translation, not a symmetry: it says nothing unless $R ≅ R^{op}$ . A property is side-neutral only when it can be restated using two-sided data — ideals, units, nilpotence — and the existence of a faithful simple left module admits no such restatement.

Compare the Jacobson radical, whose element-wise test is two-sided: $y \in rad R$ iff $1 - x y z$ is a unit for all $x, z$ . No analogous unit-theoretic characterisation of left primitivity exists, and $(11.28)$ shows why: the criterion mixes a left ideal with the lattice of two-sided ideals, so it cannot be made side-free.

Where an example must live

Three earlier results in §11 eliminate almost everything. A left artinian prime ring is simple, and simple rings are primitive on both sides. A commutative primitive ring is a field. A prime ring with a minimal left ideal is left and right primitive, and its faithful simple modules are unique up to isomorphism. Since every left primitive ring is prime, a one-sided example survives only outside all three.

Left primitive, not right primitive — what such a ring must be
- Forced properties
  - prime and semiprimitive
  - noncommutative
  - not left artinian and not right artinian
  - zero left socle and zero right socle
  - not isomorphic to its own opposite ring
- Permitted
  - domains and non-domains alike
  - algebras over any field
  - rings with infinitely many non-isomorphic faithful simple left modules

The engine: comaximality

The workable form of left primitivity is $(11.28)$ : find a proper left ideal that is comaximal with every nonzero two-sided ideal. Constructions then proceed by listing the nonzero elements of the ring and, for each one, adding a generator to the left ideal that forces comaximality — while proving that the accumulated left ideal never becomes the whole ring. Formanek's proof of $(11.29)$ is exactly that, executed with a length function on a free product.

Key Results

Lemma(11.28)Comaximality criterion for left primitivity

A ring $R$ is left primitive if and only if there exists a proper left ideal $𝔄 ⊊ R$ that is comaximal with every nonzero two-sided ideal $𝔅$ of $R$ , that is, $𝔄 + 𝔅 = R$ for all $𝔅 \neq 0$ .

Proof

**( $\Leftarrow$ )** Suppose such an $𝔄$ exists. By Zorn's Lemma enlarge $𝔄$ to a maximal left ideal $𝔪 \supseteq 𝔄$ ; comaximality is inherited, since $𝔪 + 𝔅 \supseteq 𝔄 + 𝔅 = R$ . The module $M = R / 𝔪$ is simple, and $ann (M)$ is a two-sided ideal contained in $𝔪$ . If $ann (M) \neq 0$ then comaximality would give $𝔪 + ann (M) = R$ , whereas the left-hand side equals $𝔪 \neq R$ . Hence $ann (M) = 0$ , so $M$ is a faithful simple left module and $R$ is left primitive.

**( $\Rightarrow$ )** Suppose $R$ is left primitive and let $M$ be a faithful simple left module. Writing $M ≅ R / 𝔪$ for a maximal left ideal $𝔪$ , take $𝔄 = 𝔪$ , which is proper. Let $𝔅 \neq 0$ be a two-sided ideal. If $𝔅 \subseteq 𝔪$ then $𝔅 M = 𝔅 R / 𝔪 \subseteq 𝔪 / 𝔪 = 0$ , contradicting faithfulness. So $𝔅 not \subseteq 𝔪$ , and maximality of $𝔪$ among left ideals forces $𝔪 + 𝔅 = R$ .

Proposition—Constraints on a one-sided example

Let $R$ be left primitive but not right primitive. Then $R$ is prime and semiprimitive; $R$ is not commutative; $R$ is neither left nor right artinian; and both socles vanish, $soc (_{R} R) = soc (R_{R}) = 0$ .

Proof

Primeness and semiprimitivity hold for every left primitive ring, by $(11.6)$ . If $R$ were commutative, $(11.8)$ would make it a field, hence right primitive.

Socles. Suppose $R$ has a minimal left ideal. Since $R$ is prime, $(11.11)$ says that primeness, left primitivity and right primitivity are equivalent for such a ring; so $R$ would be right primitive. Hence there are no minimal left ideals and $soc (_{R} R) = 0$ . The same argument on the other side — using that $R$ is prime and applying the right-handed form of $(11.11)$ — rules out minimal right ideals, so $soc (R_{R}) = 0$ as well.

Chain conditions. If $R$ were left artinian, then being prime it would be simple by $(11.7)$ , and a simple ring is both left and right primitive by $(11.6)$ . If $R$ were right artinian, the right-handed version of $(11.7)$ applies to the prime ring $R$ and again yields simplicity. Either way $R$ would be right primitive, a contradiction.

Proposition(11.23)Samuel: the free algebra on two generators is left primitive

Let $k$ be a field and $V = ⨁_{i \geq 1} e_{i} k$ . Define $f, g \in E = End (V_{k})$ by

f (e_{1}) = 0, f (e_{i}) = e_{i - 1} (i \geq 2), g (e_{i}) = e_{i^{2} + 1} (i \geq 1) .

Then the $k$ -algebra homomorphism $D : k ⟨ x, y ⟩ \to E$ with $D (x) = f$ , $D (y) = g$ is injective, its image acts irreducibly on $V$ , and consequently the free algebra $k ⟨ x, y ⟩$ is a left primitive ring.

Proof

Irreducibility. Let $0 \neq W \subseteq V$ be a submodule for the image $R = D (k ⟨ x, y ⟩)$ , and choose $0 \neq w \in W$ whose expression $w = e_{i_{1}} a_{1} + \dots + e_{i_{n}} a_{n}$ ( $i_{1} < \dots < i_{n}$ , all $a_{j} \neq 0$ ) has the fewest terms. Applying $f^{i_{1}}$ annihilates the first term and shortens the expression, so minimality forces $n = 1$ ; thus some $e_{i_{1}} \in W$ , and then $f^{i_{1} - 1} e_{i_{1}} = e_{1} \in W$ . Since $g^{m} e_{1} = e_{r (m)}$ with $r (m) \to \infty$ , applying a large power of $g$ and then a suitable power of $f$ produces every $e_{j}$ , so $W = V$ .

Faithfulness. Each monomial $H$ in $x$ and $y$ acts on $e_{i}$ , for all sufficiently large $i$ , by $H e_{i} = e_{h (i)}$ where $h \in ℤ [t]$ is monic of degree $2^{d}$ and $d$ is the number of occurrences of $y$ in $H$ . For example $H = x y$ gives $h (t) = t^{2}$ , while $H = y x$ gives $h (t) = (t - 1)^{2} + 1$ , and $H = y^{2}$ gives $h (t) = (t^{2} + 1)^{2} + 1$ . The combinatorial claim $(11.24)$ is that distinct monomials yield distinct polynomials, proved by induction on length: monomials ending in the same letter reduce to shorter ones, and a monomial ending in $y$ produces a polynomial in even powers of $t$ only, which a monomial ending in $x$ cannot match.

Granting $(11.24)$ , let $0 \neq z = \sum_{j} a_{j} H_{j}$ with distinct monomials $H_{j}$ and $a_{j} \in k$ nonzero. For large $i$ , $z e_{i} = \sum_{j} a_{j} e_{h_{j} (i)}$ , and since the $h_{j}$ are distinct polynomials the integers $h_{j} (i)$ are eventually pairwise distinct. Hence $z e_{i} \neq 0$ , so $z$ does not act as zero and $D$ is injective.

Combining the two parts: $V$ is a simple faithful left $k ⟨ x, y ⟩$ -module, so the free algebra is left primitive.

Corollary(11.26)Free algebras on countably many generators

For any field $k$ , the free $k$ -algebra on any set of at least two and at most countably many indeterminates is left primitive. The proof uses that ${f g^{i} : i \geq 0}$ freely generates a subalgebra of $E$ which still acts irreducibly on $V$ , so each free algebra on $n + 1$ generators, $1 \leq n \leq \infty$ , appears as a ring of transformations acting faithfully and irreducibly. The hypothesis of at least two generators is essential: $k [x]$ is commutative and is primitive only if it is a field, which it is not.

Theorem(11.27)Formanek: free rings over countable domains

Let $k$ be any — not necessarily commutative — countable domain and let ${x_{i} : i \in I}$ be independent indeterminates with $| I | \geq 2$ . Then the free ring $k ⟨ x_{i} : i \in I ⟩$ is left primitive. No hypothesis on $k$ beyond countability and the absence of zero divisors is required.

Theorem(11.29)Formanek: semigroup rings of free products

Let $A, B$ be semigroups with identity, both different from ${1}$ , let $G = A * B$ be their free product, and let $k$ be a domain with $| k | \leq | G |$ . Then the semigroup ring $k G$ is left primitive, unless $| A | = | B | = 2$ , a case excluded from the statement.

Proof strategy. Assume $| A | \geq | B |$ , so $| k G | = | A |$ , and fix a bijection $a \mapsto α (a)$ between $A^{*}$ and the nonzero elements of $R = k G$ . Fix $b \in B^{*}$ and, for each $a \in A^{*}$ , define an element $β (a) \in 1 + R α (a) R$ by one of four formulas $(11.30)$ , chosen according to the type ( $A B$ , $A A$ , $B A$ or $B B$ ) of a fixed element of maximal length in the support of $α (a)$ .

The formulas are engineered so that every element of $max-supp (β (a))$ ends with $a$ or with $a b$ , and so that $max-supp (β (a))$ contains words beginning in $A$ and words beginning in $B$ . The second property guarantees that for any $0 \neq r \in R$ no cancellation occurs at maximal length in $r β (a)$ ; the first then makes the maximal supports of $r_{1} β (a_{1}), \dots, r_{m} β (a_{m})$ pairwise disjoint for distinct $a_{1}, \dots, a_{m}$ . Hence no finite sum $\sum_{i} r_{i} β (a_{i})$ can equal $1$ , so $𝔄 = \sum_{a} R β (a)$ is a proper left ideal. Since every nonzero ideal $𝔅$ contains some $α (a)$ and $β (a) \in 1 + R α (a) R \subseteq 1 + 𝔅$ , we get $1 \in 𝔄 + 𝔅$ , so $𝔄$ is comaximal with every nonzero ideal and $(11.28)$ applies.

Corollary—How (11.29) implies (11.27)

When $| I | \geq 2$ , split the indeterminates into two nonempty blocks: the free semigroup on ${x_{i} : i \in I}$ is then the free product $A * B$ of the free semigroups on the two blocks, both different from ${1}$ , and its semigroup ring over $k$ is the free ring $k ⟨ x_{i} : i \in I ⟩$ . Since $G = A * B$ is infinite, a countable domain $k$ satisfies $| k | \leq | G |$ and neither factor has order $2$ , so $(11.29)$ applies. Taking a sufficiently large index set, the same argument realises any commutative integral domain $k$ as the centre of a left primitive ring, namely $k ⟨ x_{i} : i \in I ⟩$ itself.

Remark—Free algebras are not the counterexample

Reversing words is an anti-automorphism of $k ⟨ x_{i} ⟩$ for commutative $k$ , so the free algebra is isomorphic to its own opposite ring and is therefore right primitive as well. For a noncommutative countable domain $k$ the opposite ring is $k^{op} ⟨ x_{i} ⟩$ , again covered by $(11.27)$ . These theorems supply socle-free primitive rings in abundance — the terrain on which one-sided examples are built — but the asymmetry itself must be engineered separately, as Bergman did.

Proof Techniques and Method

How these proofs work, and which moves transfer to other arguments.

Move 1

Kill every ideal one element at a time

Enumerate the nonzero elements, and for each add a generator $β$ lying in $1 + R α R$ . Comaximality with every nonzero ideal is then automatic; all the work goes into properness.

Move 2

Leading terms as an invariant

In a free product, length and reduced form make maximal supports computable. Arrange generators so their maximal supports stay disjoint after left multiplication, and a sum can never collapse to $1$ .

Move 3

Faithfulness by growth rates

Represent monomials as polynomial index maps and separate them by degree. Distinct monomials give distinct polynomials, so distinct eventual behaviour — a clean way to prove a representation of a free algebra is faithful.

Move 1 is the general recipe for building primitive rings to order, and it is the reason cardinality hypotheses appear: the enumeration of the nonzero elements must be indexable by the semigroup itself. Move 3 recurs whenever one shows a specific pair of operators generates a free algebra — the shift-and-square construction is the canonical instance.

A negative technique is equally important: to prove a ring is not right primitive one must rule out every faithful simple right module. That is a statement about all maximal right ideals at once, which is why explicit one-sided examples are hard and rare.

Worked Example

The shift algebra $k ⟨ x, y ⟩ / (x y - 1)$

Take $k$ a field, $V = ⨁_{i \geq 1} e_{i} k$ , and define $f, g \in End (V_{k})$ by

f (e_{1}) = 0, f (e_{i}) = e_{i - 1} (i \geq 2), g (e_{i}) = e_{i + 1} (i \geq 1) .

(E.1)

The backward and forward shifts.

Relations. $f g (e_{i}) = f (e_{i + 1}) = e_{i}$ for all $i$ , so $f g = 1$ . But $g f (e_{1}) = g (0) = 0$ while $g f (e_{i}) = e_{i}$ for $i \geq 2$ , so $g f = 1 - E_{11}$ where $E_{11}$ is the projection onto $e_{1} k$ . Thus $R : = k ⟨ f, g ⟩$ is a homomorphic image of $k ⟨ x, y ⟩ / (x y - 1)$ , and Lam's Exercise 11.9 shows the map is an isomorphism.

Irreducibility. Given $0 \neq W \subseteq V$ an $R$ -submodule, take $0 \neq w \in W$ with the shortest expression $w = e_{i_{1}} a_{1} + \dots + e_{i_{n}} a_{n}$ . Applying $f^{i_{1}}$ kills the first term and shortens the expression, so $n = 1$ and $e_{i_{1}} \in W$ ; then $f^{i_{1} - 1} e_{i_{1}} = e_{1} \in W$ and $g^{j - 1} e_{1} = e_{j} \in W$ for all $j$ . Hence $W = V$ and $_{R} V$ is simple; it is faithful because $R \subseteq End (V_{k})$ .

The endomorphism ring. Let $λ \in End (_{R} V)$ , written on the right. From $f (e_{1} λ) = (f e_{1}) λ = 0$ and $\ker f = e_{1} k$ we get $e_{1} λ = e_{1} a$ for some $a \in k$ . For any $j$ , $e_{j} = g^{j - 1} e_{1}$ , so

e_{j} λ = (g^{j - 1} e_{1}) λ = g^{j - 1} (e_{1} λ) = g^{j - 1} (e_{1} a) = e_{j} a,

(E.2)

so $λ$ is multiplication by $a$ and $End (_{R} V) = k$ .

By $(11.20)$ , $R$ is therefore dense in $End (V_{k})$ , and $R$ is a left primitive ring with $\dim_{k} V$ infinite, hence not left artinian.

Why this example is symmetric

Two independent checks confirm that $R$ is right primitive as well. First, $E_{11} = 1 - g f$ is a rank-one idempotent lying in $R$ , so $R$ has a minimal left ideal and nonzero socle; since $R$ is prime, $(11.11)$ makes it right primitive. Second, $x \mapsto y$ , $y \mapsto x$ induces an isomorphism $k ⟨ x, y ⟩ / (x y - 1) ≅ (k ⟨ x, y ⟩ / (x y - 1))^{op}$ , which converts a faithful simple left module into a faithful simple right module.

Frameworks and Models

The search space for a one-sided example is narrow, and it is worth having the elimination rules in one place.

Could this left primitive ring fail to be right primitive?

It has a minimal left or right idealNo. A prime ring with a minimal one-sided ideal is left and right primitive by

(11.11)

, and left primitive rings are prime. Nonzero socle closes the question immediately.

It is left or right artinianNo. A prime artinian ring is simple by

(11.7)

, and simple rings are primitive on both sides.

It admits an anti-automorphismNo. An isomorphism

R ≅ R^{op}

transports a faithful simple left module to a faithful simple right module. Free algebras and group rings fall here.

None of the abovePossibly. This is where Bergman's and Jategaonkar's examples live: socle-free, non-artinian, and not isomorphic to their opposites.

Standard primitive rings and their symmetry status
	Left primitive	Right primitive	Socle	Reason
$M_{n} (D)$	yes	yes	nonzero	simple artinian
$End (V_{k})$ , $\dim_{k} V$ infinite	yes	yes	nonzero	minimal one-sided ideals exist
Weyl algebra $A_{1}$ , characteristic $0$	yes	yes	zero	simple ring
Free algebra $k ⟨ x, y ⟩$	yes	yes	zero	word reversal is an anti-automorphism
Shift algebra $k ⟨ x, y ⟩ / (x y - 1)$	yes	yes	nonzero	contains a rank-one idempotent
Bergman's ring (or its opposite)	yes	no	zero	built to be primitive on one side only

Standard primitive rings and their symmetry status

Comparison and Classification

Which properties are side-symmetric
Property	Symmetric?	Witness or reason
$rad R$	yes	the unit test $1 - x y z \in U (R)$ is two-sided
Semiprimitive	yes	$rad R = 0$ is a statement about a two-sided ideal
Prime, semiprime	yes	defined by products of two-sided ideals
Simple	yes	defined by the two-sided ideal lattice
Primitive	no	Bergman's ring; Jategaonkar's later examples
Primitive ideal	no	apply the ring case to $R / 𝔭$
Artinian	no	$(\begin{matrix} ℚ & ℝ \\ 0 & ℝ \end{matrix})$ is right artinian, not left artinian
Noetherian	no	same triangular ring
Perfect	no	left perfect and right perfect are inequivalent
Socle	no in general	equal for semiprime rings by $(11.9)$

The pattern is legible: properties defined by two-sided ideals or by units survive the passage between sides; properties defined by the existence of one-sided modules or by one-sided chain conditions do not. Primitivity is the most important entry in the second column because it sits so close to the symmetric notion of primeness.

Relationship Map

simpleleft primitiveprimesemiprime

Every implication above is symmetric in its statement except the middle term. Reading the chain with right in place of left gives an equally valid chain, and the two middle terms are different classes of rings whose intersection contains everything with nonzero socle.

Prime ringssymmetric class

Left primitivefaithful simple left module

Left and right primitivecontains every prime ring with nonzero socle, every simple ring, and every left primitive ring admitting an anti-automorphism

Simpleno proper nonzero two-sided ideals

Simple artinian

M_{n} (D)

The set difference between the outer two bands — left primitive but not two-sided primitive — is nonempty but populated only by deliberately constructed examples.

Applications and Industry Use

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

Group rings

Primitivity of group algebras

Formanek's free-product theorem is the prototype for results asserting that a group algebra of a group with enough free structure is primitive, and hence prime with zero radical. It feeds directly into the analysis of group rings of infinite groups.

Counterexample engineering

Calibrating theorems

One-sided examples fix the exact strength of statements: any theorem asserting a symmetric conclusion from a left primitive hypothesis must be false, and any proof that appears to give one contains an error.

Free algebras

Universal constructions

Free algebras are the ambient objects for presentations of noncommutative rings. Knowing they are primitive says their radical vanishes and they are prime, which is what licenses generic constructions by generators and relations.

Semigroup algebras

Combinatorial control

The length and reduced-word structure of a free product gives an algorithmic handle on maximal supports, the same combinatorics used in rewriting systems and normal-form algorithms for finitely presented monoids.

The honest position: this material is internal to algebra. Its value is calibration — knowing which hypotheses can be weakened and which cannot — plus the supply of primitive rings that other constructions consume.

Failure Modes and Common Mistakes

Do not assume a left primitive ring has a unique faithful simple left module; uniqueness needs a minimal one-sided ideal, by $(11.11)$ .
Do not read $(11.28)$ as symmetric: $𝔄$ is a left ideal while $𝔅$ runs over two-sided ideals, and swapping the roles gives the right-handed criterion, a different condition.
Do not conclude from $rad R = 0$ that the left and right primitive ideals coincide; only their intersections do.
Do not expect a homomorphic image of a left primitive ring to be left primitive — it need not be, and Lam's Exercise 11.1 asks for a counterexample.

Historical Notes and Lessons Learned

1945Primitivity introducedJacobson defines primitive rings as those with a faithful simple module and proves the density theorem for them. The one-sided nature of the definition is noted from the outset but no example separates the sides.
1964Bergman's exampleG. Bergman constructs a ring primitive on one side and not the other, settling the question. The construction is intricate and is not reproduced in Lam's text.
late 1960sJategaonkar's examplesFurther one-sided primitive rings appear, obtained by different methods and situated inside families of principal ideal rings.
1973Formanek on free productsFormanek proves that group rings of free products are primitive under a mild cardinality hypothesis, and deduces primitivity of free rings over countable domains.
1970sFree algebras in generalThe primitivity of free algebras is extended beyond countably many indeterminates and beyond countable coefficient rings, removing the cardinality restrictions from the earlier arguments.

The lesson is about the status of unproved symmetry. For nineteen years the left-right question for primitivity was open, and the eventual answer was negative — a reminder that a definition's failure to mention a side is not evidence that the side does not matter.

Quick Reference

DefinitionLeft primitive: a faithful simple left module exists

Criterion

(11.28)

: a proper left ideal comaximal with every nonzero ideal

AsymmetryLeft primitive

not \Rightarrow

right primitive (Bergman)

ForcedAny one-sided example is prime, socle-free on both sides, non-artinian

Free algebras

k ⟨ x, y ⟩

is left primitive

(11.23)

— and right primitive too

Formanek $(11.27)$

k

a countable domain,

| I | \geq 2

\Rightarrow

k ⟨ x_{i} ⟩

left primitive

Formanek $(11.29)$

G = A * B

| k | \leq | G |

, not

| A | = | B | = 2

\Rightarrow

k G

left primitive

Radical

rad R

is the intersection of the left, and also of the right, primitive ideals

Elimination rules for one-sided candidates
If $R$ has	Then	Reference
a minimal left or right ideal	left and right primitive coincide with prime	$(11.11)$
the left or right DCC	prime forces simple, hence two-sided primitive	$(11.7)$
commutativity	primitive means field	$(11.8)$
an anti-automorphism	the two primitivities agree	$R ≅ R^{op}$
none of these	asymmetry is possible	Bergman, Jategaonkar

Frequently Asked Questions

Is there a concrete description of a left primitive ring that is not right primitive?

Bergman's construction from the mid-1960s is the original, and Jategaonkar produced others afterwards. Neither is reproduced in Lam's §11 — the text states their existence and points to the literature. What Lam does supply are the structural constraints such a ring must satisfy and the machinery used to build socle-free primitive rings.

Why must a one-sided example have zero socle?

Because a left primitive ring is prime, and a prime ring with a minimal left ideal is both left and right primitive by $(11.11)$ — the same conclusion follows from a minimal right ideal. So the presence of any minimal one-sided ideal forces symmetry, and both socles must vanish for asymmetry to be possible.

If the free algebra is left primitive, why is it not a counterexample to symmetry?

Because reversing words is an anti-automorphism of $k ⟨ x_{i} ⟩$ , so the ring is isomorphic to its opposite and the left-handed result transports to the right. The theorems of Samuel and Formanek are constructions of primitive rings, not of asymmetric ones.

Does the asymmetry affect the Jacobson radical?

No. The left primitive ideals and the right primitive ideals are genuinely different families, but both intersect to $rad R$ . This is the clearest illustration that a symmetric invariant can be assembled from asymmetric ingredients.

What is the role of the countability hypothesis in Formanek's theorem?

The construction enumerates the nonzero elements of the ring by the nonidentity elements of one free factor, so the coefficient ring must not be too large: the hypothesis in $(11.29)$ is $| k | \leq | G |$ , which for a free ring on at least two generators is implied by countability of $k$ . Later work removed the restriction for free algebras over arbitrary fields.

Can a left primitive ring have several non-isomorphic faithful simple left modules?

Yes, and this is common in the socle-free world. Lam's examples include differential and skew polynomial rings over division rings, where the isomorphism classes correspond to conjugacy classes of parameters, and free algebras, for which two explicit non-isomorphic faithful simple modules are constructed in the exercises.

References

T. Y. Lam, A First Course in Noncommutative Rings, Graduate Texts in Mathematics 131, Springer-Verlag, 1991, §11, (11.23)–(11.30) (pp. 197–202), with the remark on Bergman's example on p. 182.
G. M. Bergman, “A ring primitive on the right but not on the left”, Proceedings of the American Mathematical Society 15 (1964).
E. Formanek, “Group rings of free products are primitive”, Journal of Algebra 26 (1973).
A. V. Jategaonkar, Left Principal Ideal Rings, Lecture Notes in Mathematics 123, Springer-Verlag, 1970.
D. S. Passman, The Algebraic Structure of Group Rings, Wiley-Interscience, 1977.
L. H. Rowen, Ring Theory, Volume I, Academic Press, 1988, Chapter 2.

AI Suggested Questions

Describe Bergman's construction of a ring primitive on one side only, in as much detail as the literature allows.
Prove that a homomorphic image of a left primitive ring need not be left primitive, with an explicit example.
Show that $M_{n} (R)$ is left primitive whenever $R$ is, and decide the same question for $R [x]$ .
Verify the combinatorial claim that distinct monomials in the shift-and-square representation give distinct index polynomials.
Which group algebras of free groups are primitive, and how does that compare with Formanek's semigroup result?
Construct two non-isomorphic faithful simple left modules over the free algebra $k ⟨ x, y ⟩$ and prove they are non-isomorphic.
What is known about the case $| A | = | B | = 2$ excluded from the semigroup ring theorem?

One-Sided Primitivity

Executive Summary

Overview

Learning Objectives

Definitions

Core Concepts

Why the definition can be asymmetric at all

Where an example must live

The engine: comaximality

Key Results

Proof Techniques and Method

Kill every ideal one element at a time

Leading terms as an invariant

Faithfulness by growth rates

Worked Example

The shift algebra $k ⟨ x, y ⟩ / (x y - 1)$

Why this example is symmetric

Frameworks and Models

Comparison and Classification

Relationship Map

Applications and Industry Use

Primitivity of group algebras

Calibrating theorems

Universal constructions

Combinatorial control

Failure Modes and Common Mistakes

Historical Notes and Lessons Learned

Quick Reference

Frequently Asked Questions

References

AI Suggested Questions

Continue learning

01Executive Summary

02Overview

03Learning Objectives

04Definitions

05Core Concepts

Why the definition can be asymmetric at all

Where an example must live

The engine: comaximality

06Key Results

07Proof Techniques and Method

Kill every ideal one element at a time

Leading terms as an invariant

Faithfulness by growth rates

08Worked Example

The shift algebra k⟨x,y⟩/(xy−1)

Why this example is symmetric

09Frameworks and Models

10Comparison and Classification

11Relationship Map

12Applications and Industry Use

Primitivity of group algebras

Calibrating theorems

Universal constructions

Combinatorial control

13Failure Modes and Common Mistakes

14Historical Notes and Lessons Learned

15Quick Reference

16Frequently Asked Questions

17Related KEVOS Topics

18References

19AI Suggested Questions

Continue learning

Executive Summary

Overview

Learning Objectives

Definitions

Core Concepts

Key Results

Proof Techniques and Method

Worked Example

The shift algebra $k ⟨ x, y ⟩ / (x y - 1)$

Frameworks and Models

Comparison and Classification

Relationship Map

Applications and Industry Use

Failure Modes and Common Mistakes

Historical Notes and Lessons Learned

Quick Reference

Frequently Asked Questions

Related KEVOS Topics

References

AI Suggested Questions