The Column Ideals of a Matrix Ring: One Idempotent and (n-1) Nilpotents

Introduction

Let (D) be a division ring, let (n \geq 1), and let (A = M_n(D)). The (j)-th column ideal of (A) is the left ideal (L_j = AE_{jj}) of the matrices supported in the (j)-th column. It is a minimal left ideal, and its corner-adapted basis has a fixed shape: exactly one idempotent and (n-1) nilpotents. For (n = 2) this reads one idempotent and one nilpotent, which is the shape of the biquaternion spinor space (S = \mathbb{C}{\tilde\Pi_+,\vec\Upsilon_y} = \mathbb{B}\tilde\Pi_+); the general statement removes any appearance of coincidence from that shape.

The count is forced by the Peirce decomposition and by nothing else. The matrix units (E_{11},\dots,E_{nn}) are a complete orthogonal family of idempotents, and relative to that family the column (L_j) splits into (n) one-dimensional corners. One of them is the diagonal corner (E_{jj}L_j = E_{jj}AE_{jj}), a copy of the division ring (D), and it contains the idempotent (E_{jj}), which is the identity of that corner. The other (n-1) are the off-diagonal corners (E_{ii}AE_{jj}) with (i \neq j), and each of them is square-zero: the product of any two of its elements is zero. A nonzero square-zero element is a nilpotent and not an idempotent, so an off-diagonal corner contains no nonzero idempotent at all. A basis of the column with one element per corner therefore has exactly one idempotent and (n-1) nilpotents, and (n-1) is nothing but the number of off-diagonal corners of a column.

Two cautions belong to the statement, and both are made precise below. The count is a count of basis elements of the corner-adapted basis, and not of the idempotents of the ideal: those form an affine family (E_{jj} + N_j) of dimension (n-1) over (D), and a basis of the column can even be chosen all of whose elements are idempotents (§The Corner-Adapted Basis Is Not Every Basis). And the count needs a division ring: over a ring with a proper nonzero ideal the column need not be minimal and the corner argument does not apply (§Where the Count Does Not Apply).

The matrix units, the classification of the left ideals of (M_n(R)) and the case (n = 2) are Matrix Algebras; the Peirce decomposition and its product rule are Unital Algebras, §Idempotents and the Peirce Decomposition; ideals, idempotents and nilpotents are Rings; and the biquaternion instance is Ideals and Peirce Decompositions of the Biquaternion Algebra.

Matrix Units, Corners and the Column Ideal

The Matrix Units

Definition. For (1 \leq i, j \leq n) let (E_{ij}) be the matrix with a single (1) in position ((i,j)) and zeros elsewhere. The matrix units (E_{ij}) form a (D)-basis of (A), and their product is

$$ E_{ij}E_{kl} = \delta_{jk}\,E_{il}, \qquad E_{jj}^2 = E_{jj}, \qquad E_{ij}^2 = 0 \ \ (i \neq j), \qquad \sum_{i=1}^{n} E_{ii} = I . $$

So each diagonal unit (E_{jj}) is an idempotent and each off-diagonal unit is square-zero, and (\dim_D A = n^2). The multiplication table, the centre and the units are those of Matrix Algebras, §Matrix Units and §Ideals and Simplicity; the units of a division ring and the vocabulary of ideals, idempotents and nilpotents are Rings.

The Column Ideal

Definition. The (j)-th column ideal of (A) is

$$ L_j = AE_{jj} = \left\{\textstyle\sum_{i=1}^{n} x_{ij}E_{ij} : x_{ij} \in D\right\}, $$

the set of matrices whose only possibly nonzero entries lie in the (j)-th column.

The name is exact: (L_j) is a left ideal, because (A \cdot L_j \subseteq AE_{jj} = L_j), and it is not a right ideal for (n \geq 2), because (E_{ij}E_{jk} = E_{ik}) carries the matrix unit (E_{ij}) of column (j) into column (k \neq j). The left ideals of (M_n(k)) over a field are exactly the sets of matrices whose columns lie in a fixed subspace, and the minimal ones are the columns, the set (L_j = I_{V_j}) with (V_j = De_j) in that article's notation; that classification is Matrix Algebras, §Ideals and Simplicity; the regular module is the direct sum (\bigoplus_j L_j) of the (n) columns, whose rank and length are Direct Sums, Free Modules and Rank.

The Corners of the Algebra

The family (E_{11},\dots,E_{nn}) is complete and pairwise orthogonal, (E_{ii}E_{kk} = \delta_{ik}E_{ii}) and (\sum_i E_{ii} = I). The Peirce decomposition of (A) with respect to it is the direct sum of the (n^2) corners

$$ A = \bigoplus_{i,k=1}^{n} E_{ii}AE_{kk}, \qquad (E_{ii}AE_{kk})(E_{ll}AE_{mm}) \subseteq \delta_{kl}\,E_{ii}AE_{mm}, $$

the corners being the Peirce spaces. A corner (E_{ii}AE_{kk}) is a subalgebra with identity (E_{ii}) when (i = k), called diagonal, and only a bimodule over the two diagonal corners when (i \neq k), called off-diagonal. The decomposition, the directness and the product rule are Unital Algebras, §Idempotents and the Peirce Decomposition.

Proposition (the corners are one-dimensional). For all (i, k),

$$ \boxed{\;E_{ii}AE_{kk} = DE_{ik}\;} . $$

Proof. Writing (X = \sum_{p,q}x_{pq}E_{pq}), the product of the two outer units is (E_{ii}XE_{kk} = \sum_{p,q}x_{pq}E_{ii}E_{pq}E_{kk} = x_{ik}E_{ik}), since (E_{ii}E_{pq} = \delta_{ip}E_{iq}) and (E_{iq}E_{kk} = \delta_{qk}E_{ik}). So a corner consists of the left multiples of one matrix unit by elements of (D).

The Corner Decomposition of a Column

The Column as a Sum of Its Corners

The column is cut by the same family, and the cut is the sum of its corners inside the column:

$$ L_j = AE_{jj} = \left(\sum_{i=1}^{n} E_{ii}\right) A E_{jj} = \bigoplus_{i=1}^{n} E_{ii}AE_{jj}, $$

using (\sum_i E_{ii} = I) and the directness of the Peirce decomposition. By §The Corners of the Algebra each summand is the line (DE_{ij}), so

$$ \boxed{\;L_j = \bigoplus_{i=1}^{n} DE_{ij}\;}, $$

a left (D)-module of dimension (n). The summand (E_{jj}AE_{jj} = DE_{jj}) is the diagonal corner of the column, and the (n-1) summands (E_{ii}AE_{jj} = DE_{ij}) with (i \neq j) are the off-diagonal corners of the column.

The Diagonal Corner

Proposition. (E_{jj}AE_{jj} = DE_{jj}) is a division ring isomorphic to (D), with identity (E_{jj}).

Proof. By §The Corners of the Algebra the corner is (DE_{jj}), and (D \to DE_{jj}), (d \mapsto dE_{jj}), is a ring isomorphism: it is additive and bijective by definition of the line, it is multiplicative because ((dE_{jj})(d'E_{jj}) = dd'E_{jj}), and it sends (1) to (E_{jj}). So the corner is a copy of the division ring (D).

The element (E_{jj}) is therefore the identity of the diagonal corner of the column, and it is the idempotent of the column's corner decomposition.

The Off-Diagonal Corners Are Square-Zero

Lemma (off-diagonal corners are square-zero). Let (e, f) be orthogonal idempotents and let (x = fae) be an element of the off-diagonal corner (fAe). Then (x^2 = 0).

Proof. (x^2 = (fae)(fae) = fa(ef)ae = 0), because (ef = 0).

The lemma is the product rule of the Peirce decomposition read at an off-diagonal pair: for (i \neq j) one has ((E_{ii}AE_{jj})^2 \subseteq E_{ii}A(E_{jj}E_{ii})AE_{jj} = 0).

Corollary. Every element of an off-diagonal corner of the column is square-zero, and hence a nilpotent; and an off-diagonal corner contains no nonzero idempotent.

Proof. The corner is (DE_{ij}) with (i \neq j) by §The Corners of the Algebra, and ((dE_{ij})^2 = d^2E_{ij}^2 = 0) for every (d), since (E_{ij}^2 = 0) when (i \neq j). A nonzero square-zero element (x) satisfies (x^2 = 0 \neq x), so it is not idempotent.

The normalisation of the basis element is irrelevant here: every nonzero element of an off-diagonal corner is square-zero, not merely the matrix unit (E_{ij}) itself. This will matter in §The Count.

The Column Is a Minimal Left Ideal

Proposition. Every nonzero element of (L_j) generates (L_j) as a left ideal, so (L_j) is a minimal left ideal of (A).

Proof. Let (0 \neq x = \sum_i x_{ij}E_{ij} \in L_j) and choose (i_0) with (x_{i_0j} \neq 0), possible because (D) has no zero divisors. For every (i),

$$ E_{ii_0}x = \sum_{k} x_{kj}E_{ii_0}E_{kj} = x_{i_0j}E_{ij}, $$

using (E_{ii_0}E_{kj} = \delta_{i_0k}E_{ij}). So every matrix unit (E_{ij}) of the column lies in the left ideal generated by (x), and with them every element of (L_j); since also (x \in L_j), that ideal is (L_j). A nonzero left ideal contained in (L_j) contains a nonzero (x) and therefore equals (L_j), so (L_j) has no nonzero proper left submodule and is minimal.

The Count

The Theorem

A (D)-basis ((x_1,\dots,x_n)) of (L_j) is corner-adapted when its (i)-th element lies in the (i)-th corner of the column,

$$ x_i \in E_{ii}L_j = E_{ii}AE_{jj} = DE_{ij} \qquad (1 \leq i \leq n). $$

Theorem (one idempotent and (n-1) nilpotents). Let ((x_1,\dots,x_n)) be a corner-adapted basis of (L_j), so that (x_i = c_iE_{ij}) with (0 \neq c_i \in D). Then:

(a) for every (i \neq j) the element (x_i = c_iE_{ij}) is square-zero, hence a nilpotent, whatever the nonzero scalar (c_i);

(b) the diagonal element (x_j = c_jE_{jj}) is idempotent if and only if (c_j = 1), that is, if and only if it is normalised to the identity (E_{jj}) of the corner;

(c) in particular the matrix-unit basis ((E_{1j},\dots,E_{nj})), the corner-adapted basis with (c_i = 1) for every (i), consists of exactly one idempotent, (E_{jj}), and (n-1) nilpotents, the off-diagonal units (E_{ij}).

Proof. (a) is the corollary of §The Off-Diagonal Corners Are Square-Zero, which holds for every nonzero scalar. (b) ((c_jE_{jj})^2 = c_j^2E_{jj}) equals (c_jE_{jj}) exactly when (c_j^2 = c_j), and in a division ring the nonzero solutions are (c_j = 1). (c) Combine (a) and (b) with the fact that (E_{jj}^2 = E_{jj}) and that a corner-adapted basis has one element per corner, of which (n-1) are off-diagonal.

So the shape of the statement is invariant to the off-diagonal normalisation and needs the diagonal normalisation: the (n-1) off-diagonal basis elements are nilpotents as they stand, and the diagonal basis element is the idempotent exactly when it is the corner's identity.

Why the Count Is Forced

The count is not a property of this or that basis element; it is the arithmetic of the column's corners. The Peirce decomposition of (L_j) has

one diagonal corner, a copy of the division ring (D), whose identity (E_{jj}) is an idempotent;
(n-1) off-diagonal corners, each square-zero, hence each devoid of nonzero idempotents.

A corner-adapted basis takes one element from each corner, so its (n-1) off-diagonal elements are nilpotents and at most one element, the diagonal one, can be idempotent; for the matrix-unit basis that element is (E_{jj}), and it is the only idempotent of that basis. Equivalently:

$$ 1 \;(\text{diagonal corner}) \;+\; (n-1) \;(\text{off-diagonal corners}) \;=\; n \;=\; \dim_D L_j , $$

where (n) is also the length of the regular module over (A), the number of its simple summands (A = \bigoplus_j L_j). The number (n-1) is thus not a numerical accident but the number of off-diagonal corners of a column.

The Nilpotents of the Column

The nilpotents of (L_j) are exactly the elements of its off-diagonal part, and each of them is square-zero. Let

$$ N_j = \bigoplus_{i \neq j} DE_{ij} \subseteq L_j $$

be the off-diagonal part of the column, of dimension (n-1).

Proposition. For (x = \sum_i x_{ij}E_{ij} \in L_j) and every (m \geq 1), the coefficient of (E_{ij}) in (x^m) is (x_{ij}x_{jj}^{m-1}); consequently:

(a) (x^2 = 0) if and only if (x_{jj} = 0), that is, if and only if (x \in N_j) (the element (0) included);

(b) the nilpotent elements of (L_j) are exactly the elements of (N_j), and each of them is square-zero;

(c) (N_j^2 = 0).

Proof. Write (x^m = \sum_i c_i^{(m)}E_{ij}). From ((uv){ij} = u) vanishes unless (m = j) in the sum ((uv)}v_{jj}) for (u, v \in L_j), which follows because (u_{im{ij} = \sum_m u \neq 0), so no power of a nonzero such (x) is zero. (c) Apply (a) to every element of (N_j).}v_{mj}), the recursion (c_i^{(m+1)} = c_i^{(m)}x_{jj}) with (c_i^{(1)} = x_{ij}) gives (c_i^{(m)} = x_{ij}x_{jj}^{m-1}). (a) For (m = 2): (x^2 = 0) means (x_{ij}x_{jj} = 0) for every (i). If (x_{jj} = 0) this holds; if (x_{jj} \neq 0) then (x_{ij} = 0) for every (i), since (D) has no zero divisors, so (x = 0). (b) By (a), a nonzero (x) with (x_{jj} = 0) has (x^2 = 0), so it is nilpotent; and if (x_{jj} \neq 0) then every power has coefficient (x_{ij}x_{jj}^{m-1}), which vanishes for no (m) when some (x_{ij

Corollary (the diagonal coefficient is a ring homomorphism). The map

$$ \pi : L_j \to D, \qquad \pi(x) = x_{jj}, $$

is a surjective ring homomorphism with kernel (N_j) and (\pi(E_{jj}) = 1), so (L_j/N_j \cong D) and (N_j) is a square-zero two-sided ideal of (L_j), viewed as a ring under the inherited product. That ring has right identity (E_{jj}), since (xE_{jj} = x) for every (x \in L_j), and it has no two-sided identity for (n \geq 2), since (E_{jj}x = x_{jj}E_{jj}) fails to be (x) as soon as the diagonal coefficient of (x) is not (1).

Proof. The map is additive and (\pi(xy) = (xy){jj} = x) = 1). Its kernel is the set of column elements with vanishing diagonal coefficient, which is (N_j) by definition.}y_{jj} = \pi(x)\pi(y)), and (\pi(E_{jj

The Idempotents of the Column

The idempotents of (L_j) are not the single element (E_{jj}); they are an affine family.

Proposition. An element (x \in L_j) is idempotent if and only if (x = 0) or (x = E_{jj} + \nu) for some (\nu \in N_j). Equivalently, the idempotents of the column are the origin together with the affine space (E_{jj} + N_j), of dimension (n-1) over (D).

Proof. Let (x = \sum_i x_{ij}E_{ij}) satisfy (x^2 = x). By §The Nilpotents of the Column the coefficient of (E_{ij}) in (x^2) is (x_{ij}x_{jj}), so the equation reads (x_{ij}x_{jj} = x_{ij}) for every (i), and for (i = j), (x_{jj}^2 = x_{jj}), so (x_{jj} \in {0,1}) because (D) is a division ring. If (x_{jj} = 0) then (x_{ij}x_{jj} = 0 = x_{ij}) for every (i), so (x = 0). If (x_{jj} = 1) then (x = E_{jj} + \nu) with (\nu = x - E_{jj} \in N_j). Conversely, for (\nu \in N_j) one has (E_{jj}\nu = \nu_{jj}E_{jj} = 0), (\nu E_{jj} = \nu) and (\nu^2 = 0), so ((E_{jj}+\nu)^2 = E_{jj} + \nu).

The family (E_{jj} + N_j) is a translate of the nilpotent subspace, so the idempotents of the column are as numerous as the elements of (N_j \cong D^{n-1}); in the biquaternion case (n = 2) they are the idempotents (\tilde\Pi_+ + c\vec\Upsilon_y) with (c \in \mathbb{C}).

The Corner-Adapted Basis Is Not Every Basis

The count of §The Theorem is a statement about corner-adapted bases, and it should not be read as a statement about every basis of the column. Because the idempotents form the affine family (E_{jj}+N_j), a basis can be chosen from that family alone.

Proposition (an idempotent basis). For (n \geq 2) the (n) elements

$$ \boxed{\;E_{jj}, \qquad E_{jj} + E_{ij} \quad (i \neq j)\;}, $$

are idempotent and form a (D)-basis of (L_j). So a basis of the column can consist entirely of idempotents.

Proof. Each (E_{ij}) with (i \neq j) lies in (N_j), so (E_{jj}+E_{ij}) is idempotent by §The Idempotents of the Column. The (n) elements are linearly independent: their coefficients in the basis ((E_{1j},\dots,E_{nj})) are the rows of the matrix that is the identity except that its (j)-th column is filled with ones, whose determinant is (1). So they are a basis.

So "one idempotent and (n-1) nilpotents" is a statement about the corner-adapted basis, the basis with one element per Peirce corner, and it is that basis which the physical spinor statements use: the biquaternion spinor space is presented on ((\tilde\Pi_+,\vec\Upsilon_y)), not on ((\tilde\Pi_+, \tilde\Pi_+ + \vec\Upsilon_y)). What the theorem asserts is that the (n-1) off-diagonal elements of a corner-adapted basis are nilpotents for every normalisation, and that such a basis never has two idempotents, having exactly one, namely (E_{jj}), when its diagonal element is the identity of its corner.

The Counts Tabulated

One Column

The corners of a column, with what each contributes. Every entry is a count of basis elements of the corner-adapted basis.

corner of the column	number	contents	the basis element
diagonal (E_{jj}AE_{jj} = DE_{jj})	(1)	a copy of the division ring (D); the identity (E_{jj})	one idempotent, (E_{jj})
off-diagonal (E_{ii}AE_{jj} = DE_{ij}), (i \neq j)	(n-1)	a square-zero line	one nilpotent, (E_{ij})

The Whole Algebra

The same bookkeeping for the whole matrix ring reads as follows.

corners of (A = M_n(D))	number	contents
diagonal (E_{ii}AE_{ii} = DE_{ii})	(n)	copies of the division ring (D); the idempotents (E_{ii}), a complete orthogonal family
off-diagonal (E_{ii}AE_{kk} = DE_{ik}), (i \neq k)	(n(n-1))	square-zero lines; the nilpotents (E_{ik})

The two counts add up: (n + n(n-1) = n^2 = \dim_D A). Per column the count is (1 + (n-1) = n), the dimension of the column and the length of the regular module over (A). The (n) diagonal corners carry the (n) matrix-unit idempotents of the complete orthogonal family; the (n(n-1)) off-diagonal corners carry the (n(n-1)) matrix-unit nilpotents (E_{ik}) with (i \neq k). The biquaternion case (n = 2) has (2) diagonal corners and (2) off-diagonal corners, the off-diagonal ones being spanned by the nilpotents (\vec\Upsilon_x) and (\vec\Upsilon_y).

The Case (n = 2) and the Biquaternion Instance

For (n = 2) a column has one diagonal corner and one off-diagonal corner, so the count is one idempotent and one nilpotent, the smallest case in which both kinds occur. The four corners of (M_2(k)) over a field and the multiplication of the units are Matrix Algebras, §The Case (n = 2).

The corpus's worked instance is the biquaternion algebra (\mathbb{B} \cong M_2(\mathbb{C})), with (D = \mathbb{C}) and (n = 2). The first column is the spinor space

$$ S = \mathbb{B}\tilde\Pi_+ = \mathbb{C}\{\tilde\Pi_+,\vec\Upsilon_y\}, $$

with (\tilde\Pi_+ = \tfrac12(e_t+i\vec e_z)) the idempotent of the diagonal corner and (\vec\Upsilon_y = \tfrac12(i\vec e_x+\vec e_y)) the square-zero generator of the off-diagonal corner; the second column is the complement (\mathbb{B}\tilde\Pi_- = \mathbb{C}{\vec\Upsilon_x,\tilde\Pi_-}), the mirror case. The root (i\vec e_z) of (+1), ((i\vec e_z)^2=e_t), carries the two associated idempotents (\tilde\Pi_+) and (\tilde\Pi_-). The idempotents of (S) are (\tilde\Pi_+ + c\vec\Upsilon_y) with (c \in \mathbb{C}), by §The Idempotents of the Column. The ideals, the Peirce corners and the two multiplication tables are Ideals and Peirce Decompositions of the Biquaternion Algebra; the two minimal left ideals as the spinor space and its complement are The Spinor Space as a Module and as a Hilbert Space and The Complementary Space of the Spinor Space and Its Physical Meaning.

Where the Count Does Not Apply

The coefficient ring must be a division ring. The corner (E_{ii}AE_{kk} = DE_{ik}) is one-dimensional only because (D) is a division ring, and with it the corner argument. Over a ring (R) with a proper nonzero ideal (I), the column (M_n(R)E_{jj}) is not minimal: the set

$$ M_n(I)E_{jj} = \left\{\textstyle\sum_i x_{ij}E_{ij} : x_{ij} \in I\right\} $$

is a nonzero left ideal of (M_n(R)) strictly contained in (M_n(R)E_{jj}), since (E_{jj} \notin M_n(I)E_{jj}). For (R = \mathbb{Z}) and (I = 2\mathbb{Z}) it is the left ideal of the matrices whose first column has even entries and whose other columns vanish. So the one-plus-((n-1)) count is a theorem about a matrix ring over a division ring, equivalently about a simple artinian ring, and not about a matrix ring over an arbitrary coefficient ring.

The count is of a corner-adapted basis. It is not a count of the idempotents of the column, which form the affine family (E_{jj}+N_j) of §The Idempotents of the Column, and it is not a count of the ideals of the column, which is the single minimal ideal (L_j). The number (n-1) counts corners, not elements of a set of idempotents.

A trap on the scalar side. For (D) noncommutative the matrix ring carries two scalar structures and (dE_{ik} \neq E_{ik}d) in general; the article reads every corner as a set of left multiples, (E_{ii}AE_{kk} = DE_{ik}), and every coefficient (x_{ij}) as a left coefficient. For a field (D = k) the two sides coincide and the trap disappears, and the corpus's instances have (D = \mathbb{C}). The general division-ring statement is kept because the coefficient ring of a simple artinian algebra is a division ring (Wedderburn–Artin), and the count transports to that generality unchanged.

Only the diagonal idempotent persists under renormalisation. The off-diagonal nilpotency of §The Theorem(a) is insensitive to the choice of basis element in its corner, but the diagonal idempotency needs the normalisation (c_j = 1); the general basis element (c_jE_{jj}) of the diagonal corner is idempotent only for (c_j = 1). A reader who scales the diagonal basis element loses the idempotent and gets a corner-adapted basis with no idempotents at all.

Worked Verification

Corners. Every corner product (E_{ii}AE_{jj}) is the line (\mathbb{C}E_{ij}), and (AE_{jj}) is the direct sum of its (n) corners.
Square-zero. Every element of every off-diagonal corner satisfies (x^2 = 0), for (n = 2,3,4,5).
Minimality. For every nonzero (x \in L_j), and (i_0) chosen with (x_{i_0j} \neq 0), one has (E_{ii_0}x = x_{i_0j}E_{ij}), so that (Ax = L_j); the column has no nonzero proper left submodule.
Powers and nilpotents. The (E_{ij})-coefficient of (x^m) is (x_{ij}x_{jj}^{m-1}) for (m \le 4); (x) is nilpotent exactly when (x_{jj} = 0); and (N_j^2 = 0).
Idempotents. (E_{jj}+\nu) is idempotent for every (\nu \in N_j), and (E_{jj}), (E_{jj}+E_{ij}) ((i \neq j)) are (n) idempotents forming a basis of the column.
Biquaternion instance. (\tilde\Pi_+^2 = \tilde\Pi_+), (\vec\Upsilon_y^2 = 0) and (\tilde\Pi_+ + c\vec\Upsilon_y) idempotent for (100) values of (c \in \mathbb{C}).
Boundary. Over (R = \mathbb{Z}) with (I = 2\mathbb{Z}) and (n = 2), the set (M_2(I)E_{11}) is preserved by left multiplication and excludes (E_{11}), so it is a proper nonzero left ideal of (M_2(\mathbb{Z})E_{11}).

Summary

In the matrix ring (A = M_n(D)) over a division ring (D), the (j)-th column ideal (L_j = AE_{jj} = \bigoplus_i DE_{ij}) is a minimal left ideal of dimension (n), and its corner-adapted basis, the basis with one element per Peirce corner, has exactly one idempotent and (n-1) nilpotents. The count is forced by the corner structure of the column: the diagonal corner (E_{jj}AE_{jj} = DE_{jj}) is a copy of the division ring (D) and contains the idempotent (E_{jj}), the identity of that corner, while each of the (n-1) off-diagonal corners (E_{ii}AE_{jj} = DE_{ij}) is square-zero by ((fae)^2 = fa(ef)ae = 0), so it contains no nonzero idempotent and all its nonzero elements are nilpotents. Hence (1 + (n-1) = n), and (n-1) is the number of off-diagonal corners of a column, not a numerical accident.

The matrix-unit basis ((E_{1j},\dots,E_{nj})) realises the count, and the shape is stable in the off-diagonal corners under any normalisation, while the diagonal element is idempotent exactly when it is the corner's identity (E_{jj}). The nilpotents of the column are exactly the elements of its off-diagonal part (N_j), each square-zero, and (N_j) is the kernel of the ring homomorphism (\pi(x) = x_{jj}) from the column to (D), with (\pi(E_{jj}) = 1), so the column is an extension of (D) by a square-zero ideal. The idempotents, by contrast, are not the single (E_{jj}): they are the affine family (E_{jj}+N_j), and a basis of the column can be chosen all of whose elements are idempotents. The count needs a division ring: over a ring with a proper nonzero ideal the column is not minimal, and the corner argument fails. For (n = 2) the count is one idempotent and one nilpotent, which is the shape of the biquaternion spinor space (S = \mathbb{B}\tilde\Pi_+ = \mathbb{C}{\tilde\Pi_+,\vec\Upsilon_y}).

Summary of Notation

Symbol	Meaning
(D), (n), (A = M_n(D))	A division ring, an integer (n \geq 1), and the matrix ring
(E_{ij})	The matrix units, (E_{ij}E_{kl} = \delta_{jk}E_{il})
(E_{jj}), (E_{ij}) ((i \neq j))	The diagonal idempotents and the off-diagonal square-zero units
(L_j = AE_{jj})	The (j)-th column ideal, a minimal left ideal of dimension (n)
(E_{ii}AE_{kk} = DE_{ik})	The Peirce corners of the algebra, one-dimensional over (D)
(E_{jj}AE_{jj} = DE_{jj})	The diagonal corner of the column, a division ring (\cong D)
(E_{ii}AE_{jj} = DE_{ij}), (i \neq j)	The (n-1) off-diagonal corners of the column, square-zero
(N_j = \bigoplus_{i\neq j} DE_{ij})	The off-diagonal part of the column, (\dim_D N_j = n-1), (N_j^2 = 0)
(\pi(x) = x_{jj})	The diagonal-coefficient homomorphism (L_j \to D), kernel (N_j)
(E_{jj} + N_j)	The affine family of idempotents of the column
Corner-adapted basis	A basis of (L_j) with one element in each corner
(\mathbb{B} \cong M_2(\mathbb{C}))	The biquaternion instance, (n = 2), (D = \mathbb{C})
(S = \mathbb{B}\tilde\Pi_+ = \mathbb{C}{\tilde\Pi_+,\vec\Upsilon_y})	The spinor space, the first column: one idempotent, one nilpotent