Page 247 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 247
VI: Skew Lattices in Rings
Proposition 6.5.4 If R is an orthosum of ideals Qi and σ: ∑i⊕∈I Qi → R is the sum epimor-
phism, then σ: ∑⊕i∈I Qi → R is an isomorphism if and only if ∑i⊕∈I ann(Qi ) is isomorphic with
( )ann(R) under the restricted map. An isomorphism τ: ∑i⊕∈I Qi ann(Qi ) → R/ann(R) is defined by
τ(⟨xi + ann(Qi)⎪i ∈ I⟩) = ∑xi + ann(R) .
Proof. Since ann( ∑⊕i∈I Qi ) = ∑i⊕∈I ann(Qi ) , if σ is an isomorphism of ∑⊕i∈I Qi with R, then σ
restricts to an isomorphism, ∑i⊕∈I ann(Qi ) ≅ ann(R). Conversely, σ is at least surjective. Suppose
σ(⟨xi⎪i ∈ I⟩) = 0 in R. If some xi ≠ 0, then for some finite set of non-0 elements from distinct Qi
we have x1 + x2 + … + xn = 0. Since 0 is in ann(R), each xi lies in the annihilator of it respective
ideal. Thus if σ restricts to an isomorphism, ∑⊕i∈I ann(Qi ) ≅ ann(R), then ⟨xi⎪i ∈ I⟩ = ⟨0⎪i ∈ I⟩,
making σ an isomorphism. The final isomorphism is the one induced from the combined
epimorphism ∑⊕i∈I Qi ⎯σ⎯→ R ⎯π⎯→ R/ann(R) where π is the canonical map. £
Regarding the idempotents, we also have:
Proposition 6.5.5 If R is the orthosum of ideals Qi and σ : ∑i⊕∈I Qi → R is the sum
epimorphism, then the restriction σE: ∑i⊕∈I E(Qi ) → E(R) is a bijection of sets that is an
isomorphism of skew Boolean algebras, whenever E(R) is multiplicative. ∑i⊕∈I E(Qi ) is the subset
of ∑⊕i∈I Qi where all xi ∈E(Qi). It equals E( ∑i⊕∈I Qi ) and is multiplicative iff each E(Qi) is.)
Proof. Since any sum of mutually orthogonal idempotents is also idempotent, σE is at least well-
defined. Let e ∈ E(R) equal x1 +...+ xr, with each xi ∈ Qi. Then e = e2 = x12 +...+ xr2 and thus for
each i ≤ r, xi2 = xi + ai where ai ∈ ann(R). Again one has xi4 = xi2 so that each xi2 ∈ E(Qi) and σE
is at least surjective. Let e ∈ E(R) be represented as both e1 +...+ er and f1 +...+ fr where
ei, fi ∈ E(Qi). (By letting some values be 0 we may assume a common indexing.) But then
ei = fi + ai where ai ∈ann(R) for each index i. Squaring both sides gives ei = fi. Thus σE is a
bijection. Finally, since σ is a ring homomorphism, the final assertions are clear. £
Given a ring R, E(R) satisfies the descending chain condition (the DCC) if any sequence
e1 ≥ e2 ≥ e3 ≥ . . .
in E(R) eventually stabilizes: en = en+1 = . . . .
The ascending chain condition on E(R) (the ACC) is defined in dual fashion. The latter implies
the former since a descending chain e1 ≥ e2 ≥ e3 ≥ . . . in E(R) induces a corresponding ascending
chain of idempotents e1 − e2 ≤ e1 – e3 ≤ . . . with both stabilizing, if they do, simultaneously.
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Proposition 6.5.4 If R is an orthosum of ideals Qi and σ: ∑i⊕∈I Qi → R is the sum epimor-
phism, then σ: ∑⊕i∈I Qi → R is an isomorphism if and only if ∑i⊕∈I ann(Qi ) is isomorphic with
( )ann(R) under the restricted map. An isomorphism τ: ∑i⊕∈I Qi ann(Qi ) → R/ann(R) is defined by
τ(⟨xi + ann(Qi)⎪i ∈ I⟩) = ∑xi + ann(R) .
Proof. Since ann( ∑⊕i∈I Qi ) = ∑i⊕∈I ann(Qi ) , if σ is an isomorphism of ∑⊕i∈I Qi with R, then σ
restricts to an isomorphism, ∑i⊕∈I ann(Qi ) ≅ ann(R). Conversely, σ is at least surjective. Suppose
σ(⟨xi⎪i ∈ I⟩) = 0 in R. If some xi ≠ 0, then for some finite set of non-0 elements from distinct Qi
we have x1 + x2 + … + xn = 0. Since 0 is in ann(R), each xi lies in the annihilator of it respective
ideal. Thus if σ restricts to an isomorphism, ∑⊕i∈I ann(Qi ) ≅ ann(R), then ⟨xi⎪i ∈ I⟩ = ⟨0⎪i ∈ I⟩,
making σ an isomorphism. The final isomorphism is the one induced from the combined
epimorphism ∑⊕i∈I Qi ⎯σ⎯→ R ⎯π⎯→ R/ann(R) where π is the canonical map. £
Regarding the idempotents, we also have:
Proposition 6.5.5 If R is the orthosum of ideals Qi and σ : ∑i⊕∈I Qi → R is the sum
epimorphism, then the restriction σE: ∑i⊕∈I E(Qi ) → E(R) is a bijection of sets that is an
isomorphism of skew Boolean algebras, whenever E(R) is multiplicative. ∑i⊕∈I E(Qi ) is the subset
of ∑⊕i∈I Qi where all xi ∈E(Qi). It equals E( ∑i⊕∈I Qi ) and is multiplicative iff each E(Qi) is.)
Proof. Since any sum of mutually orthogonal idempotents is also idempotent, σE is at least well-
defined. Let e ∈ E(R) equal x1 +...+ xr, with each xi ∈ Qi. Then e = e2 = x12 +...+ xr2 and thus for
each i ≤ r, xi2 = xi + ai where ai ∈ ann(R). Again one has xi4 = xi2 so that each xi2 ∈ E(Qi) and σE
is at least surjective. Let e ∈ E(R) be represented as both e1 +...+ er and f1 +...+ fr where
ei, fi ∈ E(Qi). (By letting some values be 0 we may assume a common indexing.) But then
ei = fi + ai where ai ∈ann(R) for each index i. Squaring both sides gives ei = fi. Thus σE is a
bijection. Finally, since σ is a ring homomorphism, the final assertions are clear. £
Given a ring R, E(R) satisfies the descending chain condition (the DCC) if any sequence
e1 ≥ e2 ≥ e3 ≥ . . .
in E(R) eventually stabilizes: en = en+1 = . . . .
The ascending chain condition on E(R) (the ACC) is defined in dual fashion. The latter implies
the former since a descending chain e1 ≥ e2 ≥ e3 ≥ . . . in E(R) induces a corresponding ascending
chain of idempotents e1 − e2 ≤ e1 – e3 ≤ . . . with both stabilizing, if they do, simultaneously.
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