Page 248 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 248
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
In what follows by a rectangular ring we mean an idempotent-dominated ring R for which E(R)
is a 0-rectangular band.
Theorem 6.5.6 Given an idempotent-closed and dominated ring R for which E(R)
satisfies the descending chain condition:
i) E(R) = ∑i⊕∈I Pi where the Pi are the primitive bands given by the union the
minimal nonzero D-classes of E(R) with {0}.
ii) R is an orthosum ideals ∑i∈I Qi where each ideal Qi is a rectangular subring for
which E(Qi) = Pi.
iii) R/ann(R) ≅ ∑i⊕∈I Qi ann(Qi ) where annihilators of all quotient rings vanish.
iv) In particular, R ≅ ∑i⊕∈I Qi when ann(R) vanishes.
v) As skew Boolean algebras, E(R) ≅ E[R/ann(R)] and Pi ≅ E[Qi/ann(Qi)].
Proof. (iii) through (v) follow from (i) and (ii) and Results 6.4.6 and 6.5.3 - 6.5.5. The DCC on
E(R) plus the normality of E(R) guarantee that each idempotent e > 0 is a unique sum of primitive
idempotents, e = p1 + ··· + pn, where each pi covers 0 in (E(R), ≤) and p1, ···, pn are mutually
orthogonal, coming from different primitive subalgebras of E(R). Assertion (i) follows from this.
Next let x ∈ Γ(R) be given. By Theorem 6.4.3.
x = exe = (p1 + ··· + pn)x(p1 + ··· + pn) = ∑ i, j pi xp j = ∑ i pixpi
for the appropriate primitive idempotents, where pixpj = 0 for i ≠ j thanks to Theorem 6.5.3. Thus
x = exe = ∑ i pi xpi where pixpi ∈ Qi and (pixpi)(pjxpj) = 0 for i ≠ j. Since every element in R is a
finite sum of elements in Γ(R), (ii) follows. £
Corollary 6.5.7 The conclusions of Theorem 6.5.6 hold if we assume the ascending
chain condition on E(R). In this case the number of summands Qi is finite, equaling the number
of atoms in E(R)/D. Conversely, when only finitely many summands Qi exist, E(R) satisfies the
ascending chain condition.
Proof. The DCC must hold on E(R) also. The ACC also prevents E(R) from having an infinite
number of 0-minimal D-classes and thus R from having an infinite number of ortho-summands Qi.
The converse is clear. £
Corollary 6.5.8 If R satisfies the conditions of Theorem 6.5.6, then KR = ∑KQi. Upon
∑choosing a nonzero idempotent ei ∈ Qi for each i, A = i eiQiei is an idempotent-covered
abelian ring such that R = A ⊕ KR as additive groups and R/KR ≅ A as rings. £
When the DCC holds on E(R), the question of E(R) being multiplicative can be reduced
as follows. To begin, let M(R) denote the set of primitive idempotents of E(R) and let M0(R)
246
In what follows by a rectangular ring we mean an idempotent-dominated ring R for which E(R)
is a 0-rectangular band.
Theorem 6.5.6 Given an idempotent-closed and dominated ring R for which E(R)
satisfies the descending chain condition:
i) E(R) = ∑i⊕∈I Pi where the Pi are the primitive bands given by the union the
minimal nonzero D-classes of E(R) with {0}.
ii) R is an orthosum ideals ∑i∈I Qi where each ideal Qi is a rectangular subring for
which E(Qi) = Pi.
iii) R/ann(R) ≅ ∑i⊕∈I Qi ann(Qi ) where annihilators of all quotient rings vanish.
iv) In particular, R ≅ ∑i⊕∈I Qi when ann(R) vanishes.
v) As skew Boolean algebras, E(R) ≅ E[R/ann(R)] and Pi ≅ E[Qi/ann(Qi)].
Proof. (iii) through (v) follow from (i) and (ii) and Results 6.4.6 and 6.5.3 - 6.5.5. The DCC on
E(R) plus the normality of E(R) guarantee that each idempotent e > 0 is a unique sum of primitive
idempotents, e = p1 + ··· + pn, where each pi covers 0 in (E(R), ≤) and p1, ···, pn are mutually
orthogonal, coming from different primitive subalgebras of E(R). Assertion (i) follows from this.
Next let x ∈ Γ(R) be given. By Theorem 6.4.3.
x = exe = (p1 + ··· + pn)x(p1 + ··· + pn) = ∑ i, j pi xp j = ∑ i pixpi
for the appropriate primitive idempotents, where pixpj = 0 for i ≠ j thanks to Theorem 6.5.3. Thus
x = exe = ∑ i pi xpi where pixpi ∈ Qi and (pixpi)(pjxpj) = 0 for i ≠ j. Since every element in R is a
finite sum of elements in Γ(R), (ii) follows. £
Corollary 6.5.7 The conclusions of Theorem 6.5.6 hold if we assume the ascending
chain condition on E(R). In this case the number of summands Qi is finite, equaling the number
of atoms in E(R)/D. Conversely, when only finitely many summands Qi exist, E(R) satisfies the
ascending chain condition.
Proof. The DCC must hold on E(R) also. The ACC also prevents E(R) from having an infinite
number of 0-minimal D-classes and thus R from having an infinite number of ortho-summands Qi.
The converse is clear. £
Corollary 6.5.8 If R satisfies the conditions of Theorem 6.5.6, then KR = ∑KQi. Upon
∑choosing a nonzero idempotent ei ∈ Qi for each i, A = i eiQiei is an idempotent-covered
abelian ring such that R = A ⊕ KR as additive groups and R/KR ≅ A as rings. £
When the DCC holds on E(R), the question of E(R) being multiplicative can be reduced
as follows. To begin, let M(R) denote the set of primitive idempotents of E(R) and let M0(R)
246
