Page 255 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 255
VI: Skew Lattices in Rings
⎡ V11 V12 V1n ⎤
⎢ ⎥
⎢ V21 V22 V2n ⎥
V = ⎢ ⎥
⎢ ⎥
⎢⎣ V1n V2n Vnn ⎦⎥
in R. If Vij ≠ 0 for some i ≠ j, then (Ei + EiVEj) ∈ E(R) but (Ei + EiVEj)Ej = EiVEj ∉ E(R) and R
is not idempotent-closed. Thus under this basis R is indeed similar to a ring in the stated block
from with each block ring Ri ≅ {EiVE⎪V ∈R}. The atomic nature of the Ei insures that each
E(Ri) is as stated. Given the diagonal block design, the final assertion is clear. £
R is isomorphically a direct sum ⊕iRi. Internally,
R = E1RE1 ⊕ ... ⊕ EkREk.
What is more,
E(R) = E(E1RE1) ⊕ ··· ⊕ E(EkREk) = {E1, 0}⊕ ··· ⊕{Ek, 0}
in that each idempotent in R decomposes uniquely as a sum of idempotents in each EiREi.
The general case: the idempotents
To pass to idempotent-closed subrings of Mn(F) in general, we first look at bands
and skew lattices in matrix rings. As it turns out: all bands in Mn(F) are simultaneously similar
to a band of upper-triangular matrices. Consequently, each skew lattice in Mn(F) is
simultaneously similar to a skew lattice of upper-triangular matrices. The result for bands was
proved for algebraically closed fields by Radjavi [1997]. The arbitrary case for bands follows
from Okninski’s results [1997]. For maximal normal bands, i.e. maximal skew Boolean algebras,
results of Cvetko-Vah ([2005b] and [2007]) are relevant. We summarize her results for
convenience.
253
⎡ V11 V12 V1n ⎤
⎢ ⎥
⎢ V21 V22 V2n ⎥
V = ⎢ ⎥
⎢ ⎥
⎢⎣ V1n V2n Vnn ⎦⎥
in R. If Vij ≠ 0 for some i ≠ j, then (Ei + EiVEj) ∈ E(R) but (Ei + EiVEj)Ej = EiVEj ∉ E(R) and R
is not idempotent-closed. Thus under this basis R is indeed similar to a ring in the stated block
from with each block ring Ri ≅ {EiVE⎪V ∈R}. The atomic nature of the Ei insures that each
E(Ri) is as stated. Given the diagonal block design, the final assertion is clear. £
R is isomorphically a direct sum ⊕iRi. Internally,
R = E1RE1 ⊕ ... ⊕ EkREk.
What is more,
E(R) = E(E1RE1) ⊕ ··· ⊕ E(EkREk) = {E1, 0}⊕ ··· ⊕{Ek, 0}
in that each idempotent in R decomposes uniquely as a sum of idempotents in each EiREi.
The general case: the idempotents
To pass to idempotent-closed subrings of Mn(F) in general, we first look at bands
and skew lattices in matrix rings. As it turns out: all bands in Mn(F) are simultaneously similar
to a band of upper-triangular matrices. Consequently, each skew lattice in Mn(F) is
simultaneously similar to a skew lattice of upper-triangular matrices. The result for bands was
proved for algebraically closed fields by Radjavi [1997]. The arbitrary case for bands follows
from Okninski’s results [1997]. For maximal normal bands, i.e. maximal skew Boolean algebras,
results of Cvetko-Vah ([2005b] and [2007]) are relevant. We summarize her results for
convenience.
253
