Ideals of Matrix Rings and the Correspondence Theorem

Q: Does the theorem need R to be commutative, or artinian, or anything?

No. The only hypotheses are that R has an identity and n is a finite positive integer. No commutativity, no chain condition, no finiteness of R. That generality is what makes (3.1) usable as the first step of the Wedderburn programme, where the base ring is an arbitrary division ring.

Q: Is the correspondence canonical, or does it depend on the choice of matrix units?

The bijection itself is canonical: I (1,1)-entries and A M_n(A) are defined without choices. The *proof* uses matrix units, and any other set of matrix units would do — a fact worth noting, because in an abstractly given ring the matrix units are not canonical while the ideal correspondence is.

Q: How do I find the ideal generated by a given matrix?

Take the two-sided ideal A of R generated by all n^2 entries; the answer is M_n(A). This makes the ideal generated by a single matrix surprisingly large: in M_2(Z), a matrix with two coprime entries generates the whole ring.

Q: What is the analogue for infinite matrices?

It fails. For an infinite-dimensional right D-vector space V, the ring E = End(V_D) has the finite-rank endomorphisms as a proper nonzero ideal even though D is simple. Lam exhibits E/I as a simple non-artinian ring precisely by exploiting this; more generally the ideals of E are indexed by infinite cardinals ≤ _D V.

Executive Summary

The two-sided ideal theory of a full matrix ring is as simple as it could possibly be: it is a carbon copy of the ideal theory of the base ring. If $R$ is a ring with identity and $n \geq 1$ , every ideal of $M_{n} (R)$ has the form $M_{n} (𝔄)$ for exactly one ideal $𝔄 ⊴ R$ , and the assignment $𝔄 \mapsto M_{n} (𝔄)$ is an isomorphism of lattices.

This is Lam's $(3.1)$ , and it is the first step of the Wedderburn programme: it produces the simple rings $M_{n} (D)$ out of the division rings $D$ , which is where the whole classification starts.

$M_{n} (𝔄)$ Shape of every ideal

1–1Ideal lattices of

R

and

M_{n} (R)

(3.1)Lam's numbering

Two-sidedOnly; one-sided fails

Overview

Passing from $R$ to $M_{n} (R)$ changes a great deal. The ring stops being commutative, acquires zero divisors and idempotents, and its one-sided ideal theory becomes genuinely richer. What does not change is the two-sided ideal theory.

{𝔄 : 𝔄 ⊴ R} ⟷ {I : I ⊴ M_{n} (R)}, 𝔄 ⟼ M_{n} (𝔄),

(3.1)

An inclusion-preserving bijection with inverse $I \mapsto {(1,1)-entries of members of I}$ .

The bijection is order-preserving in both directions, so it carries intersections to intersections, sums to sums, products to products, and maximal ideals to maximal ideals. Everything definable from the two-sided ideal lattice alone — simplicity, primeness, semiprimeness — therefore transfers verbatim between $R$ and $M_{n} (R)$ . This is the first visible instance of Morita invariance, long before Morita theory is available.

The immediate payoff is that $M_{n} (D)$ is simple for every division ring $D$ — the input to Matrix Rings over Division Rings: The Model Simple Artinian Ring and, through it, to the Wedderburn–Artin Theorem.

Learning Objectives

State $(3.1)$ with the hypotheses actually used: $R$ has an identity, $n$ is finite.
Derive the matrix-unit identity $E_{i j} M E_{k l} = m_{j k} E_{i l}$ from $E_{i j} E_{k l} = δ_{j k} E_{i l}$ .
Prove both inclusions $I \subseteq M_{n} (𝔄)$ and $M_{n} (𝔄) \subseteq I$ .
Deduce simplicity of $M_{n} (R)$ from simplicity of $R$ , and the converse.
Identify $M_{n} (R) / M_{n} (𝔄) ≅ M_{n} (R / 𝔄)$ and $Z (M_{n} (R)) ≅ Z (R)$ .
Exhibit a left ideal of $M_{2} (k)$ that is not $M_{2} (𝔏)$ for any left ideal $𝔏$ .

Definitions

Definition(3.0)Matrix units

Let $R$ be a ring with identity and $n \geq 1$ . For $1 \leq i, j \leq n$ , $E_{i j} \in M_{n} (R)$ denotes the matrix with $1$ in position $(i, j)$ and $0$ elsewhere. These matrix units form a free left $R$ -basis of $M_{n} (R)$ , so every $M = (m_{i j})$ is written uniquely as $M = \sum_{i, j} m_{i j} E_{i j}$ , and they multiply by

E_{i j} E_{k l} = δ_{j k} E_{i l}, \sum_{i = 1}^{n} E_{i i} = 1 .

$M_{n} (𝔄)$: For $𝔄 ⊴ R$ , the set of matrices with all entries in $𝔄$ . It is an ideal of $M_{n} (R)$ , and it has an identity only when $𝔄 = R$ .
$⊴$: Denotes a two-sided ideal throughout this page. One-sided ideals are always named as such.
Simple ring: A nonzero ring whose only two-sided ideals are $0$ and itself. No chain condition is assumed.
Ideal lattice: The set of two-sided ideals ordered by inclusion, with meet $\cap$ and join $+$ .
$Z (R)$: The centre of $R$ : the elements commuting with everything.

All rings have an identity and n is a finite positive integer. Both hypotheses are used; see Failure Modes.

Core Concepts

The matrix-unit calculus

Fix $M = (m_{i j}) \in M_{n} (R)$ . Multiplying on the left by $E_{i j}$ lifts row $j$ of $M$ into row $i$ and deletes everything else; multiplying on the right by $E_{k l}$ selects column $k$ and places it in column $l$ . Doing both leaves exactly one surviving entry.

E_{i j} M E_{k l} = m_{j k} E_{i l}

(3.2)

The extraction identity. Note the index shuffle: the entry that survives is $(j, k)$ , and it lands in position $(i, l)$ .

Two special cases carry the whole proof of $(3.1)$ . Taking $i = l = 1$ gives $E_{1 j} M E_{k 1} = m_{j k} E_{11}$ , which reads off an arbitrary entry of $M$ into the corner. Taking $j = k = 1$ gives $E_{i 1} M E_{1 l} = m_{11} E_{i l}$ , which plants the corner entry of $M$ into an arbitrary position.

M \in I

m_{j k} E_{11} \in I

(read off)

m_{j k} \in 𝔄

m_{j k} E_{i l} \in I

(plant back)

Why the corner-entry set is an ideal of the base

Given $I ⊴ M_{n} (R)$ , set $𝔄 = {m_{11} : (m_{i j}) \in I}$ . Additivity is inherited from $I$ . For $r \in R$ and $M \in I$ , the product $(r E_{11}) M E_{11}$ lies in $I$ and equals $r m_{11} E_{11}$ , so $r 𝔄 \subseteq 𝔄$ ; symmetrically $E_{11} M (r E_{11}) = m_{11} r E_{11}$ gives $𝔄 r \subseteq 𝔄$ . So $𝔄$ is two-sided — and note that this step consumes the two-sidedness of $I$ , which is precisely what a one-sided ideal cannot supply.

What the correspondence preserves

Because both directions of the bijection are visibly inclusion-preserving, it is an isomorphism of partially ordered sets and therefore of lattices. Concretely $M_{n} (𝔄) \cap M_{n} (𝔅) = M_{n} (𝔄 \cap 𝔅)$ , $M_{n} (𝔄) + M_{n} (𝔅) = M_{n} (𝔄 + 𝔅)$ and $M_{n} (𝔄) \cdot M_{n} (𝔅) = M_{n} (𝔄 𝔅)$ ; the last uses that a product of two matrices over $𝔄$ and $𝔅$ has entries that are sums of products, together with $n \geq 1$ to realise every such sum.

Key Results

Theorem(3.1)Ideals of a full matrix ring

Let $R$ be a ring with identity and let $n \geq 1$ . Then every two-sided ideal $I$ of $M_{n} (R)$ has the form $I = M_{n} (𝔄)$ for a uniquely determined two-sided ideal $𝔄$ of $R$ , namely $𝔄 = {m_{11} : (m_{i j}) \in I}$ . Consequently, if $R$ is a simple ring then so is $M_{n} (R)$ .

Proof

The map is well defined. If $𝔄 ⊴ R$ then $M_{n} (𝔄)$ is an additive subgroup, and for $A \in M_{n} (𝔄)$ , $B \in M_{n} (R)$ the entries of $B A$ and $A B$ are finite sums $\sum_{k} b_{i k} a_{k j}$ and $\sum_{k} a_{i k} b_{k j}$ , each term in $𝔄$ . So $M_{n} (𝔄) ⊴ M_{n} (R)$ .

Injectivity. $𝔄$ is recovered from $M_{n} (𝔄)$ as its set of $(1, 1)$ -entries, so $M_{n} (𝔄) = M_{n} (𝔅)$ forces $𝔄 = 𝔅$ .

Surjectivity. Let $I ⊴ M_{n} (R)$ and let $𝔄$ be its set of $(1, 1)$ -entries, an ideal of $R$ by the computation in Core Concepts.

* $I \subseteq M_{n} (𝔄)$ :* take $M = (m_{j k}) \in I$ and fix $j, k$ . By $(3.2)$ , $E_{1 j} M E_{k 1} = m_{j k} E_{11} \in I$ , so $m_{j k} \in 𝔄$ . Every entry of every member of $I$ lies in $𝔄$ .

* $M_{n} (𝔄) \subseteq I$ :* let $A = (a_{i l}) \in M_{n} (𝔄)$ . Since $A = \sum_{i, l} a_{i l} E_{i l}$ and $I$ is additively closed, it suffices to show $a E_{i l} \in I$ for each $a \in 𝔄$ and each pair $(i, l)$ . Choose $M \in I$ with $m_{11} = a$ . By $(3.2)$ with $j = k = 1$ , $E_{i 1} M E_{1 l} = m_{11} E_{i l} = a E_{i l} \in I$ .

Simplicity. If the only ideals of $R$ are $0$ and $R$ , the only ideals of $M_{n} (R)$ are $M_{n} (0) = 0$ and $M_{n} (R)$ ; and $M_{n} (R) \neq 0$ because $R \neq 0$ .

Corollary(3.1a)Simplicity is exactly reflected

For a ring $R$ with identity and $n \geq 1$ : $M_{n} (R)$ is simple iff $R$ is simple. In particular $M_{n} (D)$ is simple for every division ring $D$ .

Proof

( $\Leftarrow$ ) is the last sentence of $(3.1)$ . ( $\Rightarrow$ ) If $0 \neq 𝔄 ⊊ R$ were an ideal, then $0 \neq M_{n} (𝔄) ⊊ M_{n} (R)$ would be a proper nonzero ideal, contradicting simplicity. For $D$ a division ring, the only ideals are $0$ and $D$ : any nonzero ideal contains a unit.

Corollary(3.1b)Quotients commute with matrices

For $𝔄 ⊴ R$ , entrywise reduction induces a ring isomorphism

M_{n} (R) / M_{n} (𝔄) ≅ M_{n} (R / 𝔄) .

Hence $M_{n} (𝔄)$ is a maximal ideal of $M_{n} (R)$ exactly when $𝔄$ is a maximal ideal of $R$ , and a prime (respectively semiprime) ideal exactly when $𝔄$ is.

Proof

The surjection $M_{n} (R) \to M_{n} (R / 𝔄)$ reducing each entry modulo $𝔄$ is a ring homomorphism because the formulas for matrix sum and product are polynomial in the entries; its kernel is the set of matrices with all entries in $𝔄$ , that is $M_{n} (𝔄)$ . The statements about maximality and primeness follow because $M_{n} (S)$ is simple iff $S$ is simple (Corollary $(3.1 a)$ ), and $M_{n} (S)$ is a prime (semiprime) ring iff $S$ is.

Proposition(3.1c)Centre of a matrix ring

For any ring $R$ with identity and any $n \geq 1$ , $Z (M_{n} (R)) = {c \cdot 1_{n} : c \in Z (R)} ≅ Z (R)$ , where $1_{n}$ is the identity matrix.

Proof

Let $A = (a_{i j})$ be central. Comparing $A E_{k l}$ and $E_{k l} A$ using $(3.2)$ : $A E_{k l}$ has column $l$ equal to column $k$ of $A$ and all other columns zero, while $E_{k l} A$ has row $k$ equal to row $l$ of $A$ and all other rows zero. Equality forces $a_{i k} = 0$ for $i \neq k$ and $a_{k k} = a_{l l}$ for all $k, l$ ; so $A = c \cdot 1_{n}$ . Commuting with $r \cdot 1_{n}$ for all $r \in R$ then forces $c \in Z (R)$ . Conversely every such matrix is clearly central.

Remark—Radicals also transfer

The same style of argument, carried out in Lam's radical chapter, gives $rad M_{n} (R) = M_{n} (rad R)$ for every ring $R$ . That statement is not a formal consequence of $(3.1)$ , because $rad$ is not defined from the two-sided ideal lattice alone; but it fits the same pattern and is proved with the same matrix units.

Proof Techniques and Method

How this proof works, and which move to reuse.

Move 1

Read off, then plant back

To show an ideal is determined by one entry, use $E_{1 j} M E_{k 1}$ to move an arbitrary entry into the corner, and $E_{i 1} M E_{1 l}$ to move the corner back out. Two applications of one identity.

Move 2

Recover the invariant, get injectivity free

Whenever a construction $𝔄 \mapsto F (𝔄)$ admits an explicit formula recovering $𝔄$ from $F (𝔄)$ , injectivity and uniqueness are immediate. Here the formula is 'take $(1, 1)$ -entries'.

Move 3

Additive generators suffice

An ideal is an additive subgroup, so to contain $M_{n} (𝔄)$ it is enough to contain each $a E_{i l}$ . Reducing to rank-one generators is the standard economy in matrix arguments.

Move 1 generalises well beyond matrices: it is the model for arguments in rings containing a set of matrix units, that is, elements $e_{i j}$ with $e_{i j} e_{k l} = δ_{j k} e_{i l}$ and $\sum e_{i i} = 1$ . Any such ring is isomorphic to $M_{n} (S)$ with $S$ the centraliser of the $e_{i j}$ , and the ideal correspondence follows for it too.

Worked Example

Ideals of $M_{2} (ℤ)$

The ideals of $ℤ$ are $m ℤ$ for $m \geq 0$ , so by $(3.1)$ the ideals of $M_{2} (ℤ)$ are exactly $M_{2} (m ℤ)$ , one for each $m \geq 0$ , totally ordered by divisibility reversed. There are no others — in particular no ideal consisting of, say, the matrices of even determinant.

Take the ideal generated by a single matrix. Since $(3.2)$ shows the ideal generated by $M$ contains every $m_{j k} E_{i l}$ , the ideal generated by $M$ is $M_{n} (𝔄)$ with $𝔄$ the ideal of $R$ generated by all entries of $M$ . So in $M_{2} (ℤ)$ ,

⟨ (\begin{matrix} 2 & 3 \\ 0 & 0 \end{matrix}) ⟩ = M_{2} (2 ℤ + 3 ℤ) = M_{2} (ℤ),

(E.1)

A visibly non-invertible matrix generates the unit ideal, because $\gcd (2, 3) = 1$ .

By $(3.1 b)$ , $M_{2} (ℤ) / M_{2} (p ℤ) ≅ M_{2} (𝔽_{p})$ , which is simple; so $M_{2} (p ℤ)$ is a maximal ideal, matching the maximality of $p ℤ$ .

Counting: $M_{3} (ℤ / 12)$

The ideals of $ℤ / 12$ are $(d) / (12)$ for the divisors $d \in {1, 2, 3, 4, 6, 12}$ — six of them. Hence $M_{3} (ℤ / 12)$ has exactly six ideals, regardless of the matrix size $3$ . Its unique maximal ideals are $M_{3} ((2))$ and $M_{3} ((3))$ , with simple quotients $M_{3} (𝔽_{2})$ and $M_{3} (𝔽_{3})$ .

A left ideal that is not of matrix form

Let $k$ be a field and $𝔏 = {(\begin{matrix} a & 0 \\ b & 0 \end{matrix}) : a, b \in k} \subseteq M_{2} (k)$ , the matrices whose second column vanishes. Left multiplication acts columnwise, so $𝔏$ is a left ideal; it is nonzero and proper. But $M_{2} (k)$ is simple, so the only sets of the form $M_{2} (𝔄)$ are $0$ and $M_{2} (k)$ . Hence $𝔏$ is not $M_{2}$ of anything, and $(3.1)$ is genuinely a statement about two-sided ideals.

Comparison and Classification

Which properties pass between $R$ and $M_{n} (R)$ ( $n \geq 2$ )
Property of $R$	Holds for $M_{n} (R)$ ?	Reason
Simple	yes, and conversely	$(3.1)$ — lattice isomorphism
Prime	yes, and conversely	primeness is a condition on the ideal lattice
Semiprime	yes, and conversely	same
Left artinian	yes, and conversely	$M_{n} (R) ≅ End$ of a f.g. free module
Left noetherian	yes, and conversely	same argument with ACC
Left primitive	yes, and conversely	Morita invariant; needs the density theorem
Commutative	no	$E_{12} E_{21} \neq E_{21} E_{12}$
Domain	no	$E_{12}^{2} = 0$
Local	no	$E_{11}$ is a nontrivial idempotent
Division ring	no	$M_{n} (D)$ has zero divisors for $n \geq 2$

Which ideal-type questions the correspondence settles
	Two-sided	Left	Right	Subring
Determined by base ring data	yes	no	no	no
Lattice isomorphic to that of $R$	yes	no	no	no
Closed under $𝔄 \mapsto M_{n} (𝔄)$	yes	yes	yes	yes
Every such object of that form	yes	no	no	no

Which ideal-type questions the correspondence settles

Relationship Map

$(3.1)$ is a hinge: it converts a fact about division rings into a supply of simple rings, and it is used again in the uniqueness half of Wedderburn–Artin.

$(3.1)$ Ideal correspondence — $𝔄 \leftrightarrow M_{n} (𝔄)$
- immediately gives
  - $M_{n} (D)$ is simple for $D$ a division ring
  - $M_{n} (R) / M_{n} (𝔄) ≅ M_{n} (R / 𝔄)$
  - maximal, prime and semiprime ideals correspond
- feeds into
  - Matrix Rings over Division Rings: The Model Simple Artinian Ring, part (1)
  - The Wedderburn–Artin Theorem, existence half
  - Uniqueness in the Wedderburn–Artin Decomposition, via the simple components
- anticipates
  - Morita invariance of the ideal lattice
  - $rad M_{n} (R) = M_{n} (rad R)$

D

division ring

M_{n} (D)

simple

M_{n} (D)

simple artinianbuilding block of every semisimple ring

Applications and Industry Use

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

Morita theory

The first invariance statement

$R$ and $M_{n} (R)$ are Morita equivalent, and $(3.1)$ is the shadow of that equivalence on ideal lattices. In practice one proves a property is Morita invariant by checking it survives passage to $M_{n} (R)$ and to corner rings $e R e$ .

Operator algebras

Matrix amplification

For a $C^{*}$ -algebra $A$ , the closed ideals of $M_{n} (A)$ are $M_{n} (J)$ for closed ideals $J ⊴ A$ . Matrix amplification is why complete positivity, rather than positivity, is the right morphism condition in that subject.

Coding theory

Codes over matrix alphabets

Codes taking values in $M_{n} (𝔽_{q})$ that are required to be two-sided ideals collapse to codes over $𝔽_{q}$ by $(3.1)$ ; useful structure appears only if one asks for one-sided ideals, which is exactly the case the theorem excludes.

Symbolic computation

Normalising presentations

Recognising a set of matrix units inside an algebra lets a computer algebra system rewrite it as $M_{n} (S)$ and work with the smaller ring $S$ . This is the standard preprocessing step before a Wedderburn decomposition.

Honestly stated: $(3.1)$ is infrastructure. It is used constantly and cited rarely, because once you know it you stop noticing that you are using it.

Standards and Notation

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

Preferred

M_{n} (R)

(Lam, Rowen, Jacobson)

Variants

{Mat}_{n} (R)

R^{n \times n}

M_{n} R

Matrix units

E_{i j}

here;

e_{i j}

in most representation-theory sources

Ideal symbol

𝔄 ⊴ R

for two-sided; Lam uses fraktur letters for ideals throughout

MarkupPresentation MathML per ISO/IEC 40314; matrix delimiters per ISO 80000-2

GAP / MagmaMatrixAlgebra, MatrixRing; ideals via TwoSidedIdeal

Computational Notes

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

Collect entriesGiven generators

M^{(1)}, \dots, M^{(t)}

of an ideal of

M_{n} (R)

, gather all

t n^{2}

entries.

Generate in the baseCompute

𝔄 = ⟨ all those entries ⟩ ⊴ R

. Over

ℤ

this is a

\gcd

; over

k [x_{1}, \dots, x_{d}]

a Gröbner basis; over a general noncommutative

R

it is the two-sided ideal generated by them.

Return the matrix idealThe ideal generated is

M_{n} (𝔄)

, by

(3.2)

. Membership testing for

A \in M_{n} (𝔄)

n^{2}

independent membership tests in

R

Quotient if neededWork in

M_{n} (R / 𝔄)

rather than in

M_{n} (R) / M_{n} (𝔄)

— the same ring by

(3.1 b)

, but the former has cheaper arithmetic.

The reduction is a genuine complexity win: ideal arithmetic in $M_{n} (R)$ costs the same as in $R$ , not $n^{2}$ times as much.
GAP, Magma and Sage all represent matrix algebras over a base and exploit this; ideal membership is delegated to the base ring's routines.
Recognising a set of $n^{2}$ matrix units inside an abstractly presented algebra is the hard direction and is done by idempotent lifting plus a Peirce decomposition, not by $(3.1)$ .

Failure Modes and Common Mistakes

$M_{n} (𝔄)$ is an ideal, hence a ring without identity unless $𝔄 = R$ ; do not treat it as a matrix ring over a ring with $1$ .
The correspondence is about ideals, not about subrings: $M_{n} (R)$ has many subrings — upper triangular matrices, scalar matrices, $M_{n / 2} (R)$ blocks — with no counterpart in $R$ .
Infinite matrix rings are outside the theorem. The ring of column-finite $ℕ \times ℕ$ matrices over a field has a proper nonzero ideal (the finite-rank matrices) even though the field is simple.
$Z (M_{n} (R)) ≅ Z (R)$ , not $M_{n} (Z (R))$ — a slip that makes dimension counts in Wedderburn theory come out wrong by a factor of $n^{2}$ .

Quick Reference

TheoremEvery

I ⊴ M_{n} (R)

M_{n} (𝔄)

𝔄

unique

Recovery

𝔄 = {(1, 1) -entries of I}

Key identity

E_{i j} M E_{k l} = m_{j k} E_{i l}

Simplicity

M_{n} (R)

simple

\Leftrightarrow

R

simple

Quotients

M_{n} (R) / M_{n} (𝔄) ≅ M_{n} (R / 𝔄)

Centre

Z (M_{n} (R)) ≅ Z (R)

Generated ideal

⟨ M ⟩ = M_{n} (⟨ entries of M ⟩)

Not coveredone-sided ideals, subrings, infinite matrices, rngs

Worked instances at a glance
Ring	Number of ideals	Maximal ideals
$M_{n} (D)$ , $D$ a division ring	2	$0$
$M_{2} (ℤ)$	one per $m \geq 0$	$M_{2} (p ℤ)$ , $p$ prime
$M_{3} (ℤ / 12)$	6	$M_{3} ((2))$ , $M_{3} ((3))$
$M_{n} (k [x])$ , $k$ a field	one per monic $f$ and $0$	$M_{n} ((f))$ , $f$ irreducible
$M_{n} (R_{1} \times R_{2})$	product of the two counts	$M_{n} (𝔪 \times R_{2})$ and mirror

Frequently Asked Questions

Does the theorem need R to be commutative, or artinian, or anything?

No. The only hypotheses are that $R$ has an identity and $n$ is a finite positive integer. No commutativity, no chain condition, no finiteness of $R$ . That generality is what makes $(3.1)$ usable as the first step of the Wedderburn programme, where the base ring is an arbitrary division ring.

Why does the same statement fail for left ideals?

Because the proof needs to multiply on both sides. Extracting the $(j, k)$ -entry uses $E_{1 j} M E_{k 1}$ ; a left ideal only allows the left factor. Structurally, the left ideals of $M_{n} (R)$ correspond to submodules of the free module $R^{n}$ rather than to left ideals of $R$ , and for $R = D$ a division ring that is a projective space of minimal left ideals versus the two left ideals of $D$ .

Is the correspondence canonical, or does it depend on the choice of matrix units?

The bijection itself is canonical: $I \mapsto {(1, 1) -entries}$ and $𝔄 \mapsto M_{n} (𝔄)$ are defined without choices. The proof uses matrix units, and any other set of matrix units would do — a fact worth noting, because in an abstractly given ring the matrix units are not canonical while the ideal correspondence is.

How do I find the ideal generated by a given matrix?

Take the two-sided ideal $𝔄$ of $R$ generated by all $n^{2}$ entries; the answer is $M_{n} (𝔄)$ . This makes the ideal generated by a single matrix surprisingly large: in $M_{2} (ℤ)$ , a matrix with two coprime entries generates the whole ring.

Does

(3.1)

say

R

and

M_{n} (R)

are the same ring in some sense?

They are Morita equivalent, which means their module categories are equivalent, and consequently everything defined from the module category — the ideal lattice, the radical, simplicity, primeness, primitivity, global dimension — matches. They are not isomorphic for $n \geq 2$ : $M_{n} (R)$ is never commutative and always has nontrivial idempotents.

What is the analogue for infinite matrices?

It fails. For an infinite-dimensional right $D$ -vector space $V$ , the ring $E = End (V_{D})$ has the finite-rank endomorphisms as a proper nonzero ideal even though $D$ is simple. Lam exhibits $E / I$ as a simple non-artinian ring precisely by exploiting this; more generally the ideals of $E$ are indexed by infinite cardinals $\leq \dim_{D} V$ .

References

T. Y. Lam, A First Course in Noncommutative Rings, Graduate Texts in Mathematics 131, Springer-Verlag, 1991, §3, (3.1)–(3.2) (pp. 31–32).
T. Y. Lam, Lectures on Modules and Rings, Graduate Texts in Mathematics 189, Springer-Verlag, 1999, §17–§18 (matrix rings, Morita theory).
N. Jacobson, Structure of Rings, American Mathematical Society Colloquium Publications 37, revised edition, 1964, Chapter III.
F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, 2nd edition, Graduate Texts in Mathematics 13, Springer-Verlag, 1992, §17 and §22.
L. H. Rowen, Ring Theory, Volume I, Academic Press, 1988, §1.1 and §2.1.

AI Suggested Questions

Prove that a ring containing a full set of $n^{2}$ matrix units is isomorphic to $M_{n} (S)$ for $S$ the centraliser of those units.
Classify the left ideals of $M_{n} (D)$ for $D$ a division ring, and count the minimal ones when $D = 𝔽_{q}$ .
Show that $rad M_{n} (R) = M_{n} (rad R)$ using only matrix units.
Describe the ideals of the ring of column-finite $ℕ \times ℕ$ matrices over a field.
Which properties of rings are Morita invariant, and which fail — give a property preserved by $R \mapsto M_{n} (R)$ but not by Morita equivalence in general.
Give an example of rings $R \neq S$ with $M_{2} (R) ≅ M_{2} (S)$ .
How does the ideal correspondence interact with the centre when $R$ is an algebra over a field?

01Executive Summary

02Overview

03Learning Objectives

04Definitions

05Core Concepts

The matrix-unit calculus

Why the corner-entry set is an ideal of the base

What the correspondence preserves

06Key Results

07Proof Techniques and Method

Read off, then plant back

Recover the invariant, get injectivity free

Additive generators suffice

08Worked Example

Ideals of M2(ℤ)

Counting: M3(ℤ/12)

A left ideal that is not of matrix form

09Comparison and Classification

10Relationship Map

11Applications and Industry Use

The first invariance statement

Matrix amplification

Codes over matrix alphabets

Normalising presentations

12Standards and Notation

13Computational Notes

14Failure Modes and Common Mistakes

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

Ideals of $M_{2} (ℤ)$

Counting: $M_{3} (ℤ / 12)$

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