Page 249 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 249
VI: Skew Lattices in Rings
denote M(R) ∪ {0}. If E(R) satisfies this chain condition, then for any e > 0 in E(R) an m ∈ M(R)
exists such that e ≥ m. A result of Dolˇzan [8] for a case when R is abelian can be extended:
Theorem 6.5.9 If E(R) satisfies the descending chain condition, then E(R) is
multiplicative if and only if M0(R) is multiplicative.
Proof. Let M0(R) be multiplicative and let S consist of all possible finite sums ∑ei of elements
from distinct D-classes in M0(R). Since all products ef from distinct D-classes in M0(R) equal 0,
S is also a set of idempotents that is closed under multiplication. Given e > 0 in E(R), let m1 ∈
M(R) be such that e ≥ m1. If e = m1, we stop. Otherwise we have e > e − m1 ≥ m2 in M(R) with
m2 orthogonal to m1 in E(R), since m1 D m2 implies m2 = m2(e – m1)m2 = m2 − m2 = 0. If e – m1
= m2, then e = m1 + m2 with m1⊥ m2 in E(R). Otherwise, e − m1 – m2 ≥ some m3 in M(R). The
DCC insures this process eventually halts to give e = m1 + · · · + mn with the mi mutually
orthogonal and thus E(R) = S. The converse is trivial. £
Although we do not use this, it can be proved that if an idempotent-closed and dominated
ring R satisfies the DCC [ACC] on (left, right) ideals, then it must satisfy the DCC [ACC] on
idempotents.
Rectangular rings
Thus to within isomorphism, the rings of the last section are direct sums of rectangular
rings ∑i⊕∈I Ri or quotient rings ( ∑⊕i∈I Ri )/I for some ideal I ⊆ ann( ∑⊕i∈I Ri ). We study these
“atomic” rings Ri with the goal of describing them in terms of rectangular bands S and rings A
with identity 1 for which E(A) = {0, 1}. Our main concern is not the precise structure of the latter
“subatomic” ring A, but rather their role in the larger “atomic” picture.
We begin with a special case that is suggestive of what occurs generally. Given a ring A
such that E(A) = {0, 1} and a rectangular band S, we form a ring A[S]. Under addition A[S] is the
free A-module on generating set S. Thus it consists of formal sums ∑ass with as ∈ A and as ≠ 0
for only finitely many s. Addition is given by: ∑ass + ∑bss = ∑(as + bs)s; multiplication is given
by distributivity subject to the constraints: (as)(bt) = (ab)(st) and 0s = 0 = ∑0s. If s ∈ S is
identified with 1s ∈ A[S], then S is a multiplicative band inside E(A[S]). But is it a maximal
rectangular band in A[S]? In what follows, at times we use just finite expressions a1s1 +...+ ansn
with ai in A and si in S, assuming that aj = 0 for all ajsj terms not showing.
247
denote M(R) ∪ {0}. If E(R) satisfies this chain condition, then for any e > 0 in E(R) an m ∈ M(R)
exists such that e ≥ m. A result of Dolˇzan [8] for a case when R is abelian can be extended:
Theorem 6.5.9 If E(R) satisfies the descending chain condition, then E(R) is
multiplicative if and only if M0(R) is multiplicative.
Proof. Let M0(R) be multiplicative and let S consist of all possible finite sums ∑ei of elements
from distinct D-classes in M0(R). Since all products ef from distinct D-classes in M0(R) equal 0,
S is also a set of idempotents that is closed under multiplication. Given e > 0 in E(R), let m1 ∈
M(R) be such that e ≥ m1. If e = m1, we stop. Otherwise we have e > e − m1 ≥ m2 in M(R) with
m2 orthogonal to m1 in E(R), since m1 D m2 implies m2 = m2(e – m1)m2 = m2 − m2 = 0. If e – m1
= m2, then e = m1 + m2 with m1⊥ m2 in E(R). Otherwise, e − m1 – m2 ≥ some m3 in M(R). The
DCC insures this process eventually halts to give e = m1 + · · · + mn with the mi mutually
orthogonal and thus E(R) = S. The converse is trivial. £
Although we do not use this, it can be proved that if an idempotent-closed and dominated
ring R satisfies the DCC [ACC] on (left, right) ideals, then it must satisfy the DCC [ACC] on
idempotents.
Rectangular rings
Thus to within isomorphism, the rings of the last section are direct sums of rectangular
rings ∑i⊕∈I Ri or quotient rings ( ∑⊕i∈I Ri )/I for some ideal I ⊆ ann( ∑⊕i∈I Ri ). We study these
“atomic” rings Ri with the goal of describing them in terms of rectangular bands S and rings A
with identity 1 for which E(A) = {0, 1}. Our main concern is not the precise structure of the latter
“subatomic” ring A, but rather their role in the larger “atomic” picture.
We begin with a special case that is suggestive of what occurs generally. Given a ring A
such that E(A) = {0, 1} and a rectangular band S, we form a ring A[S]. Under addition A[S] is the
free A-module on generating set S. Thus it consists of formal sums ∑ass with as ∈ A and as ≠ 0
for only finitely many s. Addition is given by: ∑ass + ∑bss = ∑(as + bs)s; multiplication is given
by distributivity subject to the constraints: (as)(bt) = (ab)(st) and 0s = 0 = ∑0s. If s ∈ S is
identified with 1s ∈ A[S], then S is a multiplicative band inside E(A[S]). But is it a maximal
rectangular band in A[S]? In what follows, at times we use just finite expressions a1s1 +...+ ansn
with ai in A and si in S, assuming that aj = 0 for all ajsj terms not showing.
247
