1. Elementar y Matrices
Let A b e a ring and n a n integer 1; M
n
(A) i s a free A -module wit h basis the
matrix unit s e
t
- ( 1 i , / n) whic h multipl y b y th e rul e
eijehk = ^jh eik'
In particular, if i / the n ej. = 0, so , for an y a&A, th e elemen t e? - = / + ae- i s inver-
tible. Suc h matrices ar e called elementary; they satisf y th e followin g easil y verified relations :
(Rl) 4 4
s3
*?/*'
(R2) (e
a
if, efk) = e$ i f / , /, * ar e distinct ,
(R3) (e«, e* fc) = / i f / * h an d i * k
Here the symbo l (x , y) denote s the commutato r xyx"
l y~~1Th
. e subgrou p o f GL n(A)
generated b y al l elementary matrice s will be denote d E n(A), an d calle d th e elementary
subgroup of GL n(A). Th e followin g propositio n i s immediate fro m (R2) .
PROPOSITION
(1.1)If . n3 then E
n
(A) equals its commutator subgroup
(En(A),En(A)).
If f\A - * Af i s a surjective rin g homomorphism, th e induced homomorphis m
GLn(A) -
GLn(Ar)
nee d no t b e surjectiv e (e.g. , Z - Z/5Z an d « = 1), bu t
(1) the homomorphism E n(A) En{A') is surjective
because each of th e homomorphisms e 4 -* e~ i s so.
Since th e transpos e o f e^ i s the elementar y matri x e £ i t follow s tha t
(2) ^«04 ) w steWewwdfer transposition.
If 0 = (42, ,6
n
) the n
If a ' G ^
n - 1
( 4 ) the n
«? S
e
W
(''
- C
VO a / \ 0 a
also. I n thi s way on e show s by induction o n n tha t
4
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