Page 243 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 243
VI: Skew Lattices in Rings
Returning to Example 1, blocks D, A, C and B correspond respectively to mRm, Km,
ann(R) and mK.
We briefly consider a class of rings that are always idempotent-closed. A ring is weakly
commutative if the identity xyzw = xzyw holds. Such a ring R has a nil radical NR consisting of
all nilpotent elements in R. NR is indeed an ideal and R/NR is commutative with a vanishing nil
radical. Given a commutative ring A and a normal band S, the semigroup ring A[S] is weakly
commutative. This makes idempotent-closed rings easy to find. Indeed, all examples in this
section are weakly commutative. In any idempotent-closed ring, E(R) generates a weakly
commutative subring, denoted if Q0(R).
Theorem 6.4.13 If R is a weakly commutative ring, then E(R) is multiplicative and the
subring eRe is commutative for each e ∈ E(R). The converse also holds if R is idempotent-
dominated. Finally, for any ring R, E(R) is multiplicative if and only if Q0(R) is weakly
commutative.
Proof. Given e, f ∈ E(R), (ef)2 = efef = eeff = ef. Also, given exe, eye in eRe, we have
(exe)(eye) = e(exe)(eye)e = e(eye)(exe)e = (eye)(exe).
Conversely let R be idempotent-dominated with E(R) being multiplicative and each
subring eRe, for e ∈ E(R), being commutative. Let eae, fbf, gcg, hdh in Γ(R) be given with e, f, g,
h ∈ E(R). Since E(R) is multiplicative, as in the proof of Lemma 6.4.7, e′, f′, g′, h′ in E(R) exist
such that e′ ≥ e, f′ ≥ f, g′ ≥ g, h′ ≥ h with e′, f′, g′ and h′ being D-related. Thus we may assume at
the outset that e, f, g and h are D-related. This plus the assumption that each eRe be commutative
gives
(eae)(fbf)(gcg)(hdh) = (eae)e(fbf)e(gcg)e(hdh)
= (eae)e(gcg)e(fbf)e(hdh) = (eae)(gcg)(fbf)(hdh)
holding in Γ(R). Distribution extends the identity xyzw = xzyw from Γ(R) to all of R. If we just
assume E(R) is normal, then this property extends via distribution to weak commutativity of the
generated subring Q0(R). The converse is clear. £
Idempotent-closed rings in general
What can be said about an arbitrary idempotent-closed ring R where Q(R) could be a
proper ideal? We begin by quoting a standard result in ring theory.
Lemma 6.4.14 Given a ring R with a nil ideal N, every idempotent in R/N is of the form
e + N for some idempotent e in R. £
As an immediate consequence we have:
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Returning to Example 1, blocks D, A, C and B correspond respectively to mRm, Km,
ann(R) and mK.
We briefly consider a class of rings that are always idempotent-closed. A ring is weakly
commutative if the identity xyzw = xzyw holds. Such a ring R has a nil radical NR consisting of
all nilpotent elements in R. NR is indeed an ideal and R/NR is commutative with a vanishing nil
radical. Given a commutative ring A and a normal band S, the semigroup ring A[S] is weakly
commutative. This makes idempotent-closed rings easy to find. Indeed, all examples in this
section are weakly commutative. In any idempotent-closed ring, E(R) generates a weakly
commutative subring, denoted if Q0(R).
Theorem 6.4.13 If R is a weakly commutative ring, then E(R) is multiplicative and the
subring eRe is commutative for each e ∈ E(R). The converse also holds if R is idempotent-
dominated. Finally, for any ring R, E(R) is multiplicative if and only if Q0(R) is weakly
commutative.
Proof. Given e, f ∈ E(R), (ef)2 = efef = eeff = ef. Also, given exe, eye in eRe, we have
(exe)(eye) = e(exe)(eye)e = e(eye)(exe)e = (eye)(exe).
Conversely let R be idempotent-dominated with E(R) being multiplicative and each
subring eRe, for e ∈ E(R), being commutative. Let eae, fbf, gcg, hdh in Γ(R) be given with e, f, g,
h ∈ E(R). Since E(R) is multiplicative, as in the proof of Lemma 6.4.7, e′, f′, g′, h′ in E(R) exist
such that e′ ≥ e, f′ ≥ f, g′ ≥ g, h′ ≥ h with e′, f′, g′ and h′ being D-related. Thus we may assume at
the outset that e, f, g and h are D-related. This plus the assumption that each eRe be commutative
gives
(eae)(fbf)(gcg)(hdh) = (eae)e(fbf)e(gcg)e(hdh)
= (eae)e(gcg)e(fbf)e(hdh) = (eae)(gcg)(fbf)(hdh)
holding in Γ(R). Distribution extends the identity xyzw = xzyw from Γ(R) to all of R. If we just
assume E(R) is normal, then this property extends via distribution to weak commutativity of the
generated subring Q0(R). The converse is clear. £
Idempotent-closed rings in general
What can be said about an arbitrary idempotent-closed ring R where Q(R) could be a
proper ideal? We begin by quoting a standard result in ring theory.
Lemma 6.4.14 Given a ring R with a nil ideal N, every idempotent in R/N is of the form
e + N for some idempotent e in R. £
As an immediate consequence we have:
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