JP Journal of Algebra, Number Theory and Applications
Volume 3, Issue 2, Pages 259 - 268
(August 2003)
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ADDITIVE
MAPS PRESERVING MATRIX EQUALITIES AND PARTIAL
ORDERINGS
Xiaomin Tang (P. R. China) and Chongguang Cao (P. R. China)
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Abstract: We characterize the additive
maps preserving rank-additivity on matrix
algebras. Let Mn(F) be the algebra of all n ´ n matrices over a field F. Then T
is an additive injective map preserving
rank-additivity on Mn(F) if and
only if there exist invertible matrices U, V Î Mn (F) and an injective field homomorphism
f
of F such
that either T(X)=UXfV, "X Î Mn(F) or T(X)
U(XT)fV, "X Î Mn(F),
where Xf(f(xij)).
As
applications, we determine the additive maps
preserving minus-order on Mn(F)
over
the field F. |
Keywords and phrases: additive preserver problem, rank-additivity,
minus-order. |
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