Page 235 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 235
VI: Skew Lattices in Rings
∇-bands open the door to cubic skew lattices that are distinct from any possible quadratic
skew lattice a ring. This observation, however, is tempered by the following result:
Theorem 6.3.6. Every cubic skew lattice (S: ∇, •) having a lattice section T in a ring R
is isomorphic to a quadratic skew lattice (Sʹ: ○, •) in some ring Rʹ. This always occurs when S/D
is countable and in particular for any (S: ∇, •) in a matrix ring over a (skew) field, in which case
Rʹ can also be a matrix ring.
Proof. Given a lattice section T with left and right extensions SL and SR, the internal Kimura
decomposition gives an embedding of S into a quadratic skew lattice S → SR × SL ⊆ R × R. £
6.4 Idempotent-closed rings
A ring R is idempotent-closed if its set of idempotents E(R) is multiplicatively closed.
We begin with a result in basic ring theory.
Proposition 6.4.1 Let ring R be idempotent-closed. If R has an identity, then its
idempotents commute. In general, if the idempotents of R commute, then E(R) is in the center of
R and forms a generalized Boolean algebra (E(R); ∨, ∧, \, 0) upon setting
e ∧ f = ef, e∨f = e ○ f = e + f − ef and e \ f = e − efe.
In this case, given e ∈ E(R), both eR and ann(e) = {x ∈ R⎪ex = 0} are ideals and R decomposes
internally as eR ⊕ ann(e) under the map x → ex + (x − ex).
Proof. If 1 ∈ R, then e(1−e) = 0 for all e ∈ E(R). Hence ef(1−e) = [ef(1−e)]2 = 0 and thus
ef = efe for all e, f in E(R). Similarly, fe = efe and thus ef = fe for all e, f in E(R). Next, assuming
all idempotents commute, choose e ∈ E(R) and x ∈ R. Observe first that if ex = 0, then e + xe is
idempotent; likewise, if xe = 0, then e + ex is idempotent. Thus in general, since e(x − ex) = 0
one has that e + xe − exe is idempotent. Multiplying each expression with e and using
commutation, gives e + xe − exe = e = e + ex − exe, from which xe = exe = ex follows. The
remaining assertions are easy consequences of this. £
A ring whose idempotents commute, and thus lie in the center of the ring, is called
abelian. Such rings are clearly idempotent-closed. We are less concerned here with the internal
structure of these rings, than in their role in the class of idempotent-closed rings. Suffice it to say,
under mild hypotheses abelian rings decompose into direct sums of rings with identity whose
only idempotents are 0 and 1. (See Section 6.5.)
Moving beyond the abelian case, first recall that a band S is normal if it satisfies any and
hence all of the following equivalent identities:
(a) efge = egfe. (b) efgh = egfh. (c) efege = egefe.
233
∇-bands open the door to cubic skew lattices that are distinct from any possible quadratic
skew lattice a ring. This observation, however, is tempered by the following result:
Theorem 6.3.6. Every cubic skew lattice (S: ∇, •) having a lattice section T in a ring R
is isomorphic to a quadratic skew lattice (Sʹ: ○, •) in some ring Rʹ. This always occurs when S/D
is countable and in particular for any (S: ∇, •) in a matrix ring over a (skew) field, in which case
Rʹ can also be a matrix ring.
Proof. Given a lattice section T with left and right extensions SL and SR, the internal Kimura
decomposition gives an embedding of S into a quadratic skew lattice S → SR × SL ⊆ R × R. £
6.4 Idempotent-closed rings
A ring R is idempotent-closed if its set of idempotents E(R) is multiplicatively closed.
We begin with a result in basic ring theory.
Proposition 6.4.1 Let ring R be idempotent-closed. If R has an identity, then its
idempotents commute. In general, if the idempotents of R commute, then E(R) is in the center of
R and forms a generalized Boolean algebra (E(R); ∨, ∧, \, 0) upon setting
e ∧ f = ef, e∨f = e ○ f = e + f − ef and e \ f = e − efe.
In this case, given e ∈ E(R), both eR and ann(e) = {x ∈ R⎪ex = 0} are ideals and R decomposes
internally as eR ⊕ ann(e) under the map x → ex + (x − ex).
Proof. If 1 ∈ R, then e(1−e) = 0 for all e ∈ E(R). Hence ef(1−e) = [ef(1−e)]2 = 0 and thus
ef = efe for all e, f in E(R). Similarly, fe = efe and thus ef = fe for all e, f in E(R). Next, assuming
all idempotents commute, choose e ∈ E(R) and x ∈ R. Observe first that if ex = 0, then e + xe is
idempotent; likewise, if xe = 0, then e + ex is idempotent. Thus in general, since e(x − ex) = 0
one has that e + xe − exe is idempotent. Multiplying each expression with e and using
commutation, gives e + xe − exe = e = e + ex − exe, from which xe = exe = ex follows. The
remaining assertions are easy consequences of this. £
A ring whose idempotents commute, and thus lie in the center of the ring, is called
abelian. Such rings are clearly idempotent-closed. We are less concerned here with the internal
structure of these rings, than in their role in the class of idempotent-closed rings. Suffice it to say,
under mild hypotheses abelian rings decompose into direct sums of rings with identity whose
only idempotents are 0 and 1. (See Section 6.5.)
Moving beyond the abelian case, first recall that a band S is normal if it satisfies any and
hence all of the following equivalent identities:
(a) efge = egfe. (b) efgh = egfh. (c) efege = egefe.
233
