Page 237 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 237
VI: Skew Lattices in Rings
Proof. (ii) is clear. If E(R) is multiplicative, x ∈ Γ(R) and e, f ∈ E(R) are such that ex = x = xf,
then set g = f∇e. By absorption, ge = e and fg = f. Hence gx = gex = ex = x and likewise xg = x
so that gxg = x and (ii) holds. For (iii), first, let Qʹ be the least ideal of R containing E(R). Clearly
Q(R) is a subring of Qʹ. Conversely, Qʹ consists of sums of terms of the form xe, ey and xey
where e ∈ E(R) and x, y ∈ R. For such e, x, y, observe that f = e + xe − exe and h = e + ey − eye
are both idempotent, so that xe, ey and thus xey must lie in Q(R). Indeed as defined, ef = e and
fe = f so that ff = (fe)f = f(ef) = fe = f. Similar observations hold for h. £
Elements in Γ(R) are said to be idempotent-covered. R as a whole is idempotent-covered
if its elements are thus. Rings with identity and von Neumann regular rings are trivially
idempotent-covered. For such rings we have an extension of Proposition 6.4.1.
Theorem 6.4.4 Given an idempotent-covered ring R, E(R) is multiplicative if and only if
R is abelian.
Proof. Let R be idempotent-covered and idempotent-dominated. Given e, f in E(R), by Theorem
6.4.3 g ∈E(R) exists such that g(ef − fe) = ef − fe = (ef − fe)g. By normality,
ef − fe = g(ef − fe)g = gefg − gfeg = gefg − gefg = 0.
Thus E(R) is commutative and R is abelian. The converse is clear. £
A ring R is idempotent-dominated if it is generated from the set Γ(R) of all idempotent-
covered elements, or put otherwise, if R = Q(R). Clearly idempotent-covered rings are
idempotent-dominated, but not conversely.
Idempotent-covered abelian rings enter into the general case of idempotent-closed rings
in at least two ways. First, for any idempotent e in an idempotent-closed ring R, the principal
subring eRe is abelian. Secondly, we will see that each idempotent-closed and dominated ring
has a maximal abelian image that in many cases arises as a major subring of the given ring.
Being idempotent-dominated can have a side effect. For any idempotent-covered ring R,
the annihilator ideal vanishes, ann(R) = {0}. If R is just idempotent-dominated, this need not be
the case, even when E(R) is also multiplicative. Indeed, given e, f in a multiplicatively closed
E(R), the following small rectangular band arises
efe R ef
LL
fe R fef
and the combination α(e, f) = ef + fe − efe − fef can lie in ann(R). This is of interest since for all
idempotents e and f we have e∇f = e ○ f + α(e, f). Indeed we have:
235
Proof. (ii) is clear. If E(R) is multiplicative, x ∈ Γ(R) and e, f ∈ E(R) are such that ex = x = xf,
then set g = f∇e. By absorption, ge = e and fg = f. Hence gx = gex = ex = x and likewise xg = x
so that gxg = x and (ii) holds. For (iii), first, let Qʹ be the least ideal of R containing E(R). Clearly
Q(R) is a subring of Qʹ. Conversely, Qʹ consists of sums of terms of the form xe, ey and xey
where e ∈ E(R) and x, y ∈ R. For such e, x, y, observe that f = e + xe − exe and h = e + ey − eye
are both idempotent, so that xe, ey and thus xey must lie in Q(R). Indeed as defined, ef = e and
fe = f so that ff = (fe)f = f(ef) = fe = f. Similar observations hold for h. £
Elements in Γ(R) are said to be idempotent-covered. R as a whole is idempotent-covered
if its elements are thus. Rings with identity and von Neumann regular rings are trivially
idempotent-covered. For such rings we have an extension of Proposition 6.4.1.
Theorem 6.4.4 Given an idempotent-covered ring R, E(R) is multiplicative if and only if
R is abelian.
Proof. Let R be idempotent-covered and idempotent-dominated. Given e, f in E(R), by Theorem
6.4.3 g ∈E(R) exists such that g(ef − fe) = ef − fe = (ef − fe)g. By normality,
ef − fe = g(ef − fe)g = gefg − gfeg = gefg − gefg = 0.
Thus E(R) is commutative and R is abelian. The converse is clear. £
A ring R is idempotent-dominated if it is generated from the set Γ(R) of all idempotent-
covered elements, or put otherwise, if R = Q(R). Clearly idempotent-covered rings are
idempotent-dominated, but not conversely.
Idempotent-covered abelian rings enter into the general case of idempotent-closed rings
in at least two ways. First, for any idempotent e in an idempotent-closed ring R, the principal
subring eRe is abelian. Secondly, we will see that each idempotent-closed and dominated ring
has a maximal abelian image that in many cases arises as a major subring of the given ring.
Being idempotent-dominated can have a side effect. For any idempotent-covered ring R,
the annihilator ideal vanishes, ann(R) = {0}. If R is just idempotent-dominated, this need not be
the case, even when E(R) is also multiplicative. Indeed, given e, f in a multiplicatively closed
E(R), the following small rectangular band arises
efe R ef
LL
fe R fef
and the combination α(e, f) = ef + fe − efe − fef can lie in ann(R). This is of interest since for all
idempotents e and f we have e∇f = e ○ f + α(e, f). Indeed we have:
235
