Page 253 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 253
VI: Skew Lattices in Rings
Again E(&2[S]) = {0} ∪ S, K = {0, s, a+b, c+d, a+c, b+d, a+d, b+c} so that A[S]/K ≅ &2 and
ann(&2[S]) = {0, s}. &2[S]/ann(Z2[S]) has 8 elements that are parameterized by the x-row entries.
&2[S] is weakly commutative with K equaling the nil radical N. £
6.6 Idempotent-closed rings of matrices
In this section F is again a field and n is a positive natural number. We characterize those
idempotent-closed subrings of Mn(F) that are maximal subject to certain constrains. This is done
modulo maximal idempotent-closed matrix subrings R for which E(R) ⊆ {0, I}. While we are
unaware of any general characterization of the latter, if F is the field of complex numbers 0, a
result of Livshits et al [2003] implies that if A is an algebra in Mn(0) such that E(A) ⊆ {0, I,},
then either A = N or A = 0I + N for some nil algebra N. A maximal such algebra would be
simultaneously similar to the algebra of all upper-triangular matrices with constant diagonals,
since any subring of nilpotent matrices in Mn(0) is triangularizable. (See Example 6.7.2 below.
See also Okninski [1997] or Radjavi and Rosenthal [2000].)
The abelian case
We begin with maximal idempotent-covered abelian subrings of Mn(F). Such subrings
necessarily have and identity E and the “maximal” constraint insures that E = I. The following
pair of extreme examples are suggestive of the general case.
Example 6.6.1. R is the abelian subring of all diagonal matrices in Mn(F) and E(R) is
the set of all diagonal matrices with only 0 or 1 entries. R is a maximal idempotent-closed ring in
Mn(F). For suppose that ring R′ in Mn(F) properly contains R. Let A ∈ R′ be a non-diagonal
matrix and let i ≠ j be such that EiAEj ≠ 0. (Ei is the matrix with the i-th diagonal entry 1, and 0
elsewhere.) Then (Ei + EiAEj)2 = Ei + EiAEj ∈ E(R), but (Ei + EiAEj)Ej = EiAEj is nilpotent.
Thus R′ is not idempotent-closed.
⎡ d1 0 0⎤
⎢ 0 d2 ⎥
⎢ 0 ⎥
⎢ ⎥
⎣⎢ 0 0 dn ⎥⎦
251
Again E(&2[S]) = {0} ∪ S, K = {0, s, a+b, c+d, a+c, b+d, a+d, b+c} so that A[S]/K ≅ &2 and
ann(&2[S]) = {0, s}. &2[S]/ann(Z2[S]) has 8 elements that are parameterized by the x-row entries.
&2[S] is weakly commutative with K equaling the nil radical N. £
6.6 Idempotent-closed rings of matrices
In this section F is again a field and n is a positive natural number. We characterize those
idempotent-closed subrings of Mn(F) that are maximal subject to certain constrains. This is done
modulo maximal idempotent-closed matrix subrings R for which E(R) ⊆ {0, I}. While we are
unaware of any general characterization of the latter, if F is the field of complex numbers 0, a
result of Livshits et al [2003] implies that if A is an algebra in Mn(0) such that E(A) ⊆ {0, I,},
then either A = N or A = 0I + N for some nil algebra N. A maximal such algebra would be
simultaneously similar to the algebra of all upper-triangular matrices with constant diagonals,
since any subring of nilpotent matrices in Mn(0) is triangularizable. (See Example 6.7.2 below.
See also Okninski [1997] or Radjavi and Rosenthal [2000].)
The abelian case
We begin with maximal idempotent-covered abelian subrings of Mn(F). Such subrings
necessarily have and identity E and the “maximal” constraint insures that E = I. The following
pair of extreme examples are suggestive of the general case.
Example 6.6.1. R is the abelian subring of all diagonal matrices in Mn(F) and E(R) is
the set of all diagonal matrices with only 0 or 1 entries. R is a maximal idempotent-closed ring in
Mn(F). For suppose that ring R′ in Mn(F) properly contains R. Let A ∈ R′ be a non-diagonal
matrix and let i ≠ j be such that EiAEj ≠ 0. (Ei is the matrix with the i-th diagonal entry 1, and 0
elsewhere.) Then (Ei + EiAEj)2 = Ei + EiAEj ∈ E(R), but (Ei + EiAEj)Ej = EiAEj is nilpotent.
Thus R′ is not idempotent-closed.
⎡ d1 0 0⎤
⎢ 0 d2 ⎥
⎢ 0 ⎥
⎢ ⎥
⎣⎢ 0 0 dn ⎥⎦
251
