Page 258 - Leech, Jonathan E. 2020. Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond. Koper: University of Primorska Press
P. 258
Jonathan E. Leech │ Noncommutative Lattices: Skew Lattices, Skew Boolean Algebras and Beyond
⊕ ... ⊕ EkREk by the remark after Proposition 6..7.2, given any matrix in A, all nondiagonal
blocks in its central submatrix are 0-blocks. Indeed, since P and EPE share the same central
submatrix, every matrix P in R has this pattern in its central submatrix. Hence, given our
assumptions on E(R), every matrix in R is at least of the block form (3).
⎡0 A C ⎤ ⎡0 A′ C′⎤ ⎡0 AD′ AB′ ⎤
Using simplified block-of-blocks format, ⎢0 B⎥ ⎢0 ⎥ = ⎢0 D0B′ ⎥⎥⎦
⎢⎣0 D 0 ⎥⎦ ⎢⎣0 D′ B′ ⎥⎦ ⎢⎣0 DD′
0 0 0 0
⎡ 0 A C ⎤2 ⎡0 A C⎤ ⎧⎪AD = A, DB = B, It follows that any such ring
and ⎢ 0 D B ⎥ = ⎢0 D B⎥ iff ⎨
⎩⎪AB = C and D2 = D.
⎣⎢ 0 0 0 ⎥⎦ ⎣⎢0 0 0 ⎦⎥
of matrices is idempotent-closed if and only if its central ring of D-matrices is idempotent-closed.
Thus the status of a subring R as being idempotent-closed in Mn(F) remains unchanged if its
design is extended by first allowing arbitrary Ai, Bi and C blocks and then enlarging each ring Ri
of Di-blocks to a maximal idempotent-closed subring of Mn(i)(F) that includes 0n(i) and In(i). If
such a subring extension is idempotent-covered, then the Ai, Bi, C blocks and the Ri rings already
had this maximal status as stated in the theorem.
It remains to see that the ring R of all matrices of form (3) is idempotent-dominated. We
again use this simplified block form. To begin, the identities
⎡ 0 A C ⎤ ⎡0 0 0⎤ ⎡0 A 0⎤ ⎡0 0 C⎤
⎢ 0 D B ⎥ = ⎢0 D B⎥ + ⎢0 0 0⎥ + ⎢0 0 0 ⎥ ,
⎣⎢ 0 0 0 ⎥⎦ ⎢⎣0 0 0 ⎦⎥ ⎢⎣0 0 0⎥⎦ ⎣⎢0 0 0 ⎥⎦
⎡0 0 0⎤ ⎡0 0 0⎤ ⎡0 0 0⎤ ⎡0 A 0⎤ ⎡0 A 0⎤ ⎡0 0 0⎤
⎢0 D B⎥ = ⎢0 I 0⎥ ⎢0 D B⎥ and ⎢0 0 0⎥ = ⎢0 0 0⎥ ⎢0 I 0⎥
⎣⎢0 0 0 ⎦⎥ ⎢⎣0 0 0⎥⎦ ⎣⎢0 0 0 ⎦⎥ ⎣⎢0 0 0⎦⎥ ⎢⎣0 0 0⎦⎥ ⎢⎣0 0 0⎥⎦
⎡0 0 C⎤
reduce this verification to showing any ⎢0 0⎥
⎣⎢0 0 ⎥ lies in QR. Such a matrix is a sum of matrices
0 ⎦
0
of this type having only a single nonzero entry; but any such summand easily factors into a
⎡0 A 0⎤ ⎡0 0 0⎤
product of the form ⎢0 0⎥ ⎢0 B⎥
⎢⎣0 0 0⎥⎦ ⎣⎢0 0 0 ⎦⎥ and thus lies in QR. £
0 0
A is the maximal abelian subring of R and the subring of all matrices in R for which all Di-
blocks are 0-blocks is the nilpotent ideal KR encountered in Section 4. R = A ⊕ K as additive
groups and as rings R/K ≅ A making A the maximal abelian image of R. (See Section 6.4.)
Again in our simplified block notation:
256
⊕ ... ⊕ EkREk by the remark after Proposition 6..7.2, given any matrix in A, all nondiagonal
blocks in its central submatrix are 0-blocks. Indeed, since P and EPE share the same central
submatrix, every matrix P in R has this pattern in its central submatrix. Hence, given our
assumptions on E(R), every matrix in R is at least of the block form (3).
⎡0 A C ⎤ ⎡0 A′ C′⎤ ⎡0 AD′ AB′ ⎤
Using simplified block-of-blocks format, ⎢0 B⎥ ⎢0 ⎥ = ⎢0 D0B′ ⎥⎥⎦
⎢⎣0 D 0 ⎥⎦ ⎢⎣0 D′ B′ ⎥⎦ ⎢⎣0 DD′
0 0 0 0
⎡ 0 A C ⎤2 ⎡0 A C⎤ ⎧⎪AD = A, DB = B, It follows that any such ring
and ⎢ 0 D B ⎥ = ⎢0 D B⎥ iff ⎨
⎩⎪AB = C and D2 = D.
⎣⎢ 0 0 0 ⎥⎦ ⎣⎢0 0 0 ⎦⎥
of matrices is idempotent-closed if and only if its central ring of D-matrices is idempotent-closed.
Thus the status of a subring R as being idempotent-closed in Mn(F) remains unchanged if its
design is extended by first allowing arbitrary Ai, Bi and C blocks and then enlarging each ring Ri
of Di-blocks to a maximal idempotent-closed subring of Mn(i)(F) that includes 0n(i) and In(i). If
such a subring extension is idempotent-covered, then the Ai, Bi, C blocks and the Ri rings already
had this maximal status as stated in the theorem.
It remains to see that the ring R of all matrices of form (3) is idempotent-dominated. We
again use this simplified block form. To begin, the identities
⎡ 0 A C ⎤ ⎡0 0 0⎤ ⎡0 A 0⎤ ⎡0 0 C⎤
⎢ 0 D B ⎥ = ⎢0 D B⎥ + ⎢0 0 0⎥ + ⎢0 0 0 ⎥ ,
⎣⎢ 0 0 0 ⎥⎦ ⎢⎣0 0 0 ⎦⎥ ⎢⎣0 0 0⎥⎦ ⎣⎢0 0 0 ⎥⎦
⎡0 0 0⎤ ⎡0 0 0⎤ ⎡0 0 0⎤ ⎡0 A 0⎤ ⎡0 A 0⎤ ⎡0 0 0⎤
⎢0 D B⎥ = ⎢0 I 0⎥ ⎢0 D B⎥ and ⎢0 0 0⎥ = ⎢0 0 0⎥ ⎢0 I 0⎥
⎣⎢0 0 0 ⎦⎥ ⎢⎣0 0 0⎥⎦ ⎣⎢0 0 0 ⎦⎥ ⎣⎢0 0 0⎦⎥ ⎢⎣0 0 0⎦⎥ ⎢⎣0 0 0⎥⎦
⎡0 0 C⎤
reduce this verification to showing any ⎢0 0⎥
⎣⎢0 0 ⎥ lies in QR. Such a matrix is a sum of matrices
0 ⎦
0
of this type having only a single nonzero entry; but any such summand easily factors into a
⎡0 A 0⎤ ⎡0 0 0⎤
product of the form ⎢0 0⎥ ⎢0 B⎥
⎢⎣0 0 0⎥⎦ ⎣⎢0 0 0 ⎦⎥ and thus lies in QR. £
0 0
A is the maximal abelian subring of R and the subring of all matrices in R for which all Di-
blocks are 0-blocks is the nilpotent ideal KR encountered in Section 4. R = A ⊕ K as additive
groups and as rings R/K ≅ A making A the maximal abelian image of R. (See Section 6.4.)
Again in our simplified block notation:
256
