JP Journal of Algebra, Number Theory and Applications
Volume 4, Issue 3, Pages 427 - 436
(December 2004)
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INVERSE-PRESERVING
ADDITIVE MAPS BETWEEN MATRIX SPACES OVER FIELDS
OF CHARACTERISTIC NOT 2
Lizhu Hao (P. R. China) and Xian Zhang (P. R. China)
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Abstract: Suppose F is a
field of characteristic not 2 and n is a
positive integer ³
2. Let Mn(F)
and Sn(F)
be
the n´n
full matrix space and symmetric matrix space
over F, respectively. Assume G1,
G2 Î
{Sn(F), Mn(F)}.
We say that a map f
: G1 ®
G2 is additive if f(X
+ Y)= f(X) + f(Y)
for
any X,
Y Î
G1, and preserves inverses of matrices if
f(Z–
1) = f(Z)– 1
for
every invertible Z
Î
G1. The set of all additive maps from
G1
to G2
preserving inverses of matrices is defined by A(G1,
G2). Denote A(G1)
= A
(G1, G1). In this paper the sets
A(Sn(F),
Mn(F)), A
(Sn(F)), A
(Mn(F)) and A(Mn(F),
Sn(F)) are characterized. |
Keywords and phrases: inverse-preserving,
field, additive map. |
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