Page 239 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 239
VI: Skew Lattices in Rings
Theorem 6.4.8 If R is both idempotent-dominated and idempotent-closed, then R/ann(R)
has both properties with ann(R/ann(R)) = {0} so that e∇f = e○f in R/ann(R). The natural
epimorphism π: R → R/ann(R) induces a skew Boolean algebra isomorphism
πE: E(R) ≅ E(R/ann(R)) and ring isomorphisms πE: eRe → π(e)(R/ann(R))π(e) between
corresponding principal subrings. £
When ann(R) ≠ {0}, R/ann(R) provides a cleaner, trimmer version of R, sharing many of
its characteristics, but without a non-vanishing annihilator ideal.
The canonical ideal KR
The annihilator ideal is a part of generally larger canonical nilpotent ideal KR that all
rings possess. We begin with an example.
Example 1 For n ≥ 1, Mn(F) is the ring of n × n matrices over a field F. Mn(F) is
trivially idempotent-covered since it has an identity, but E(Mn(F)) is never multiplicative unless
n = 1. For n ≥ 2, given fixed integer parameters i, j, k ≥ 0 subject to i + j + k = n and 1 ≤ j ≤ n,
consider the subring Rin, j,k of all matrices with the following block design satisfying the added
restriction that D be a diagonal matrix:
⎡ 0i×i Ai× j Ci×k ⎤
⎢ D j× j ⎥
Dijk = ⎢ 0 j×i 0k× j B j×k ⎥ .
⎢ ⎥
⎢⎣ 0k×i 0k×i k ⎥⎦
Rin, j,k is idempotent-dominated and idempotent-closed. Rin, j,k and E( Rin, j,k ) are noncommutative
when j < n and commutative when j = n. The idempotents of Rin, j,k are the matrices where D has
only 0-1 entries in the diagonal, AD = A, DB = B and AB = C (for cases where A, B or C do not
disappear.) E( Rin, j,k ) is right-handed when i = 0 and left-handed when k = 0, with C vanishing in
either case along with A or B, respectively. For i, k > 0 so that j ≤ n − 2, ann( Rin, j,k ) is nontrivial
consisting of all matrices for which blocks D, A and B are 0-submatrices. The strictly upper
triangular matrices (where D = 0) form a nilpotent ideal, K. Considering just addition, the
additive group of the ring is the direct sum of the group of all diagonal matrices (a maximal
idempotent-covered subring with central idempotents) and the group of strictly upper triangular
matrices (the nilpotent ideal K). To what extent does such a direct decomposition characterize
idempotent-dominated rings with multiplicative sets of idempotents? This leads us to the
following considerations. £
237
Theorem 6.4.8 If R is both idempotent-dominated and idempotent-closed, then R/ann(R)
has both properties with ann(R/ann(R)) = {0} so that e∇f = e○f in R/ann(R). The natural
epimorphism π: R → R/ann(R) induces a skew Boolean algebra isomorphism
πE: E(R) ≅ E(R/ann(R)) and ring isomorphisms πE: eRe → π(e)(R/ann(R))π(e) between
corresponding principal subrings. £
When ann(R) ≠ {0}, R/ann(R) provides a cleaner, trimmer version of R, sharing many of
its characteristics, but without a non-vanishing annihilator ideal.
The canonical ideal KR
The annihilator ideal is a part of generally larger canonical nilpotent ideal KR that all
rings possess. We begin with an example.
Example 1 For n ≥ 1, Mn(F) is the ring of n × n matrices over a field F. Mn(F) is
trivially idempotent-covered since it has an identity, but E(Mn(F)) is never multiplicative unless
n = 1. For n ≥ 2, given fixed integer parameters i, j, k ≥ 0 subject to i + j + k = n and 1 ≤ j ≤ n,
consider the subring Rin, j,k of all matrices with the following block design satisfying the added
restriction that D be a diagonal matrix:
⎡ 0i×i Ai× j Ci×k ⎤
⎢ D j× j ⎥
Dijk = ⎢ 0 j×i 0k× j B j×k ⎥ .
⎢ ⎥
⎢⎣ 0k×i 0k×i k ⎥⎦
Rin, j,k is idempotent-dominated and idempotent-closed. Rin, j,k and E( Rin, j,k ) are noncommutative
when j < n and commutative when j = n. The idempotents of Rin, j,k are the matrices where D has
only 0-1 entries in the diagonal, AD = A, DB = B and AB = C (for cases where A, B or C do not
disappear.) E( Rin, j,k ) is right-handed when i = 0 and left-handed when k = 0, with C vanishing in
either case along with A or B, respectively. For i, k > 0 so that j ≤ n − 2, ann( Rin, j,k ) is nontrivial
consisting of all matrices for which blocks D, A and B are 0-submatrices. The strictly upper
triangular matrices (where D = 0) form a nilpotent ideal, K. Considering just addition, the
additive group of the ring is the direct sum of the group of all diagonal matrices (a maximal
idempotent-covered subring with central idempotents) and the group of strictly upper triangular
matrices (the nilpotent ideal K). To what extent does such a direct decomposition characterize
idempotent-dominated rings with multiplicative sets of idempotents? This leads us to the
following considerations. £
237
