Page 245 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 245
VI: Skew Lattices in Rings
6.5 Decomposing E(R) and R
Recall that, every skew Boolean algebra is a subdirect product of primitive algebras,
thanks largely to the following restatement of Theorem 4.1.4.
Theorem 6.5.1 Given a D-class A of a skew Boolean algebra S, set
S1 = {e ∈ S⎪e∧a∧e = e for some (and hence all) a ∈ A},
and
S2 = {f ∈ S⎪f∧a = a∧f = 0 for some (and hence all) a ∈ A}.
Then both S1 and S2 are subalgebras of S, all elements of S1 commute with all elements of S2 and
the map µ: S1 × S2 → S defined by µ(e1, e2) = e1 ∨ e2 is an isomorphism of skew Boolean algebras.
The inverse isomorphism is given by µ−1(e) = (e∧a∧e, e \ e∧a∧e) . £
Described otherwise, S1 is the union of the D-class A and all lower D-classes in the
generalized Boolean lattice S/D, while S2 consists of all D-classes B that meet A and its lower D-
classes at {0} in S/D. In fact S1 and S2 are ideals of S where by an ideal of a skew lattice S is
meant any subset I such that e∨f ∈ I for all e, f ∈ I, and both e∧g, g∧e ∈ I for all e ∈ I and all
g ∈ S. S1 corresponds to the principal ideal in S/D determined by the element A of S/D while S2
corresponds to the ideal in S/D consisting of all elements of S/D that meet A at 0. If S = E(R),
how does this play out in the full context of the host ring, R? We begin by passing from skew
lattice ideals I in E(R) to their induced ring ideals QI in R.
Lemma 6.5.2 Let R be idempotent-closed and let I be an ideal of E(R). Then the least
ideal QI of R containing I consists of all sums ∑eixifi where both ei, fi ∈ I and xi ∈ R. (As with
Theorem 6.4.3, all elements in QI also have the form ∑eixiei where ei ∈ I.)
Proof. Since −(xy) = (−x)y = x(−y), the set of all such sums is at least a subring Rʹ of R. Clearly
I ⊆ Rʹ ⊆ QI. On the other hand, QI consists of sums of the form ∑xieiyi where ei ∈ I and xi, yi ∈ R.
But each such sum lies in Rʹ since its terms must. Indeed, given e ∈ I, f = e + xe − exe satisfies
fe = f and ef = e, forcing f to be an idempotent in I, thus ensuring xe = f – e + exe ∈ Rʹ. Similarly
ey and thus xey also lie in Rʹ. £
Theorem 6.5.3 Let ring R be idempotent-closed and dominated, and let I and J be ideals
of E(R) such that each e ∈ E(R) is uniquely f + g for some f ∈ I and g ∈ J. Then:
i) R = QI + QJ and QIQJ = {qq′ ⎢q ∈ QI & q′ ∈ QJ} = {0} = QJQI.
ii) σ: QI ⊕QJ →R given by σ(q, q′) = q + q′ is a ring homomorphism onto R, that
restricts to isomorphisms, σI : QI ⊕ {0} ≅ QI and σJ :{0} ⊕ QJ ≅ QJ.
iii) In general, QI ∩ QJ ⊆ ann(R) with QI ∩ QJ possibly exceeding {0};
iv) σ is an isomorphism when QI ∩ QJ = {0}, and in particular when ann(R) = {0}.
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6.5 Decomposing E(R) and R
Recall that, every skew Boolean algebra is a subdirect product of primitive algebras,
thanks largely to the following restatement of Theorem 4.1.4.
Theorem 6.5.1 Given a D-class A of a skew Boolean algebra S, set
S1 = {e ∈ S⎪e∧a∧e = e for some (and hence all) a ∈ A},
and
S2 = {f ∈ S⎪f∧a = a∧f = 0 for some (and hence all) a ∈ A}.
Then both S1 and S2 are subalgebras of S, all elements of S1 commute with all elements of S2 and
the map µ: S1 × S2 → S defined by µ(e1, e2) = e1 ∨ e2 is an isomorphism of skew Boolean algebras.
The inverse isomorphism is given by µ−1(e) = (e∧a∧e, e \ e∧a∧e) . £
Described otherwise, S1 is the union of the D-class A and all lower D-classes in the
generalized Boolean lattice S/D, while S2 consists of all D-classes B that meet A and its lower D-
classes at {0} in S/D. In fact S1 and S2 are ideals of S where by an ideal of a skew lattice S is
meant any subset I such that e∨f ∈ I for all e, f ∈ I, and both e∧g, g∧e ∈ I for all e ∈ I and all
g ∈ S. S1 corresponds to the principal ideal in S/D determined by the element A of S/D while S2
corresponds to the ideal in S/D consisting of all elements of S/D that meet A at 0. If S = E(R),
how does this play out in the full context of the host ring, R? We begin by passing from skew
lattice ideals I in E(R) to their induced ring ideals QI in R.
Lemma 6.5.2 Let R be idempotent-closed and let I be an ideal of E(R). Then the least
ideal QI of R containing I consists of all sums ∑eixifi where both ei, fi ∈ I and xi ∈ R. (As with
Theorem 6.4.3, all elements in QI also have the form ∑eixiei where ei ∈ I.)
Proof. Since −(xy) = (−x)y = x(−y), the set of all such sums is at least a subring Rʹ of R. Clearly
I ⊆ Rʹ ⊆ QI. On the other hand, QI consists of sums of the form ∑xieiyi where ei ∈ I and xi, yi ∈ R.
But each such sum lies in Rʹ since its terms must. Indeed, given e ∈ I, f = e + xe − exe satisfies
fe = f and ef = e, forcing f to be an idempotent in I, thus ensuring xe = f – e + exe ∈ Rʹ. Similarly
ey and thus xey also lie in Rʹ. £
Theorem 6.5.3 Let ring R be idempotent-closed and dominated, and let I and J be ideals
of E(R) such that each e ∈ E(R) is uniquely f + g for some f ∈ I and g ∈ J. Then:
i) R = QI + QJ and QIQJ = {qq′ ⎢q ∈ QI & q′ ∈ QJ} = {0} = QJQI.
ii) σ: QI ⊕QJ →R given by σ(q, q′) = q + q′ is a ring homomorphism onto R, that
restricts to isomorphisms, σI : QI ⊕ {0} ≅ QI and σJ :{0} ⊕ QJ ≅ QJ.
iii) In general, QI ∩ QJ ⊆ ann(R) with QI ∩ QJ possibly exceeding {0};
iv) σ is an isomorphism when QI ∩ QJ = {0}, and in particular when ann(R) = {0}.
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