Page 254 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 254
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
Example 6.6.2. Let R consist of all upper triangular matrices in Mn(F) with constant
diagonals. R is trivially idempotent-closed since E(R) = {0, I}.
⎡d a1, 2 a1, n ⎤
⎢ d ⎥
⎢0 a2, n ⎥
⎢ ⎥
⎢ ⎥
⎣ 0 0 d ⎦
For any field, R is maximal subject to E(R) = {0, I}. In fact, R is a maximal idempotent-closed
subring of Mn(F). For its verification, see the discussion in Example 9 in Cvetko-Vah and Leech
[2011]. That a subring has only idempotents 0 and I is unremarkable. But that a maximal
idempotent-closed subring of Mn(F) can be thus is interesting. £
Theorem 6.6.1. For each n ≥ 1, Mn(F) has a maximal idempotent-closed subring R that
is an algebra over F and for which E(R) = {0, I}. £
Conversely, one may ask: are all maximal idempotent-closed subalgebras R of Mn(F) for
which E(R) = {0, I} simultaneously similar to such an example, as happens when F = 0? In any
case, a general way of constructing maximal idempotent-closed subrings of Mn(F) with identity I
(and hence abelian) follows from the next result.
Proposition 6.6.2. An idempotent-closed ring R in Mn(F) containing I is similar
simultaneously to a ring in block form (1) below where for each index i all blocks Di form a
subring Ri in Mn(i)(F) with E(Ri) = {0n(i), In(i)}. R is maximally idempotent-closed in Mn(F) if
and only if each Ri is maximally idempotent-closed in the matrix ring Mn(i)(F).
⎡ D1 0 0⎤
⎢ 0 D2 ⎥
(1) U = ⎢ 0 ⎥
⎢ ⎥
⎢⎣ 0 0 Dn ⎦⎥
Proof. By Proposition 6.4.1, E(R) is a Boolean algebra. Let E1, ... , Ek be the atoms of E(R).
Then E1 +... + Ek = I and we can choose a basis for Fn such that in this basis each Ei is similar to
the diagonal matrix that has In(i) (of the appropriate dimension) on the i-th diagonal block entry
and 0s elsewhere. All elements of E(R) are sums of atoms and thus diagonal with the block form
(1) where each Di = In(i) or 0n(i). Given
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Example 6.6.2. Let R consist of all upper triangular matrices in Mn(F) with constant
diagonals. R is trivially idempotent-closed since E(R) = {0, I}.
⎡d a1, 2 a1, n ⎤
⎢ d ⎥
⎢0 a2, n ⎥
⎢ ⎥
⎢ ⎥
⎣ 0 0 d ⎦
For any field, R is maximal subject to E(R) = {0, I}. In fact, R is a maximal idempotent-closed
subring of Mn(F). For its verification, see the discussion in Example 9 in Cvetko-Vah and Leech
[2011]. That a subring has only idempotents 0 and I is unremarkable. But that a maximal
idempotent-closed subring of Mn(F) can be thus is interesting. £
Theorem 6.6.1. For each n ≥ 1, Mn(F) has a maximal idempotent-closed subring R that
is an algebra over F and for which E(R) = {0, I}. £
Conversely, one may ask: are all maximal idempotent-closed subalgebras R of Mn(F) for
which E(R) = {0, I} simultaneously similar to such an example, as happens when F = 0? In any
case, a general way of constructing maximal idempotent-closed subrings of Mn(F) with identity I
(and hence abelian) follows from the next result.
Proposition 6.6.2. An idempotent-closed ring R in Mn(F) containing I is similar
simultaneously to a ring in block form (1) below where for each index i all blocks Di form a
subring Ri in Mn(i)(F) with E(Ri) = {0n(i), In(i)}. R is maximally idempotent-closed in Mn(F) if
and only if each Ri is maximally idempotent-closed in the matrix ring Mn(i)(F).
⎡ D1 0 0⎤
⎢ 0 D2 ⎥
(1) U = ⎢ 0 ⎥
⎢ ⎥
⎢⎣ 0 0 Dn ⎦⎥
Proof. By Proposition 6.4.1, E(R) is a Boolean algebra. Let E1, ... , Ek be the atoms of E(R).
Then E1 +... + Ek = I and we can choose a basis for Fn such that in this basis each Ei is similar to
the diagonal matrix that has In(i) (of the appropriate dimension) on the i-th diagonal block entry
and 0s elsewhere. All elements of E(R) are sums of atoms and thus diagonal with the block form
(1) where each Di = In(i) or 0n(i). Given
252
