Page 210 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
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Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
classes must generate a ∇-band. In particular, maximal totally pre-ordered regular bands in rings
are ∇-bands.
Not only does a maximal normal band S form a skew Boolean algebra in its host ring
(with e \ f = e – efe), it is also the full set of idempotents in the subring Rʹ that it generates:
S = E(Rʹ). Conversely, if E(R) is multiplicatively closed in a given ring R, then E(R) forms a
skew Boolean algebra. (See Theorem 2.3.7 or Theorem 6.4.2 below.) In Section 4 we begin our
study of such idempotent-closed rings. Much of the focus is on the case of idempotent-dominated
rings where R = Q(R), the ideal generated from E(R). In this case, if KR is the canonical nilpotent
ideal {k ∈ R⎪xky = 0 for all x, y ∈ R}, then R/KR is the maximal abelian image of R. (See
Theorem 6.4.10. Recall that R is abelian if its idempotents commute.) When E(R) also has a
lattice section, R is a semidirect sum A ⊕ KR, that is direct under addition with A being a
maximal abelian subring of R that is necessarily isomorphic to R/KR. (Theorem 6.4.11.) This is
the case for all idempotent-closed and dominated rings of n×n matrices. When R is not
idempotent-dominated, these facts apply directly to Q(R); but upon setting K = KQ(R), then K is a
nilpotent ideal of R also, with R being idempotent-closed if and only if R/K is abelian. (Theorem
6.4.15.) These and related facts are studied in the fourth section.
Like Boolean algebras, skew Boolean algebras decompose almost at will. (See Theorem
4.1.4 or Theorem 6.5.1 below.) To what extent does this extend to idempotent-closed rings,
especially if they are idempotent-dominated? In particular, given certain finiteness conditions
(e.g. the ACC or DCC on idempotents), must ring decompose as a direct sum of subrings that in
some sense are “atomic”? These questions are pursued for idempotent-dominated rings in the
fifth section. The “atomic” rings turn out to be rectangular rings – idempotent-dominated rings
whose non-0 idempotents form a rectangular band under multiplication. Theorem 6.5.6 states
that each idempotent-closed and dominated ring R satisfying the DCC on idempotents is an
orthosum of ideals Qi (that is, R = ∑Qi with QiQj = {0} for i ≠ j) where each Qi is a rectangular
ring. While the orthosum condition is a weakening of the direct sum condition, if the annihilator
ideal of R reduces to {0}, the sum must be direct. Rectangular rings are characterized in
Theorems 6.5.11 and 6.5.14.
The results of Sections 6.4 and 6.5 are then “tested” in the context of matrix rings over
fields in Sections 6.6 and 6.7. The former studies upper triangular representations of normal
skew lattices and skew Boolean algebras in matrix rings, and Section 6.7 studies upper triangular
representations of (maximal) idempotent-closed and dominated subrings of matrix rings. (See
Theorems 6.7.4 – 6.7.6.)
The chapter ends with historical comments and relevant references.
208
classes must generate a ∇-band. In particular, maximal totally pre-ordered regular bands in rings
are ∇-bands.
Not only does a maximal normal band S form a skew Boolean algebra in its host ring
(with e \ f = e – efe), it is also the full set of idempotents in the subring Rʹ that it generates:
S = E(Rʹ). Conversely, if E(R) is multiplicatively closed in a given ring R, then E(R) forms a
skew Boolean algebra. (See Theorem 2.3.7 or Theorem 6.4.2 below.) In Section 4 we begin our
study of such idempotent-closed rings. Much of the focus is on the case of idempotent-dominated
rings where R = Q(R), the ideal generated from E(R). In this case, if KR is the canonical nilpotent
ideal {k ∈ R⎪xky = 0 for all x, y ∈ R}, then R/KR is the maximal abelian image of R. (See
Theorem 6.4.10. Recall that R is abelian if its idempotents commute.) When E(R) also has a
lattice section, R is a semidirect sum A ⊕ KR, that is direct under addition with A being a
maximal abelian subring of R that is necessarily isomorphic to R/KR. (Theorem 6.4.11.) This is
the case for all idempotent-closed and dominated rings of n×n matrices. When R is not
idempotent-dominated, these facts apply directly to Q(R); but upon setting K = KQ(R), then K is a
nilpotent ideal of R also, with R being idempotent-closed if and only if R/K is abelian. (Theorem
6.4.15.) These and related facts are studied in the fourth section.
Like Boolean algebras, skew Boolean algebras decompose almost at will. (See Theorem
4.1.4 or Theorem 6.5.1 below.) To what extent does this extend to idempotent-closed rings,
especially if they are idempotent-dominated? In particular, given certain finiteness conditions
(e.g. the ACC or DCC on idempotents), must ring decompose as a direct sum of subrings that in
some sense are “atomic”? These questions are pursued for idempotent-dominated rings in the
fifth section. The “atomic” rings turn out to be rectangular rings – idempotent-dominated rings
whose non-0 idempotents form a rectangular band under multiplication. Theorem 6.5.6 states
that each idempotent-closed and dominated ring R satisfying the DCC on idempotents is an
orthosum of ideals Qi (that is, R = ∑Qi with QiQj = {0} for i ≠ j) where each Qi is a rectangular
ring. While the orthosum condition is a weakening of the direct sum condition, if the annihilator
ideal of R reduces to {0}, the sum must be direct. Rectangular rings are characterized in
Theorems 6.5.11 and 6.5.14.
The results of Sections 6.4 and 6.5 are then “tested” in the context of matrix rings over
fields in Sections 6.6 and 6.7. The former studies upper triangular representations of normal
skew lattices and skew Boolean algebras in matrix rings, and Section 6.7 studies upper triangular
representations of (maximal) idempotent-closed and dominated subrings of matrix rings. (See
Theorems 6.7.4 – 6.7.6.)
The chapter ends with historical comments and relevant references.
208
