Idempotent elements determined matrix algebras
Let M n ( R ) be the algebra of all n × n matrices over a unital commutative ring R with 2 invertible, V be an R-module. It is shown in this article that, if a symmetric bilinear map { · , · } from M n ( R ) × M n ( R ) to V satisfies the condition that { u , u } = { e , u } whenever u 2 = u , then...
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Published in | Linear algebra and its applications Vol. 435; no. 11; pp. 2889 - 2895 |
---|---|
Main Authors | , , |
Format | Journal Article |
Language | English |
Published |
Amsterdam
Elsevier Inc
01.12.2011
Elsevier |
Subjects | |
Online Access | Get full text |
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Abstract | Let
M
n
(
R
)
be the algebra of all
n
×
n
matrices over a unital commutative ring
R with 2 invertible,
V be an
R-module. It is shown in this article that, if a symmetric bilinear map
{
·
,
·
}
from
M
n
(
R
)
×
M
n
(
R
)
to
V satisfies the condition that
{
u
,
u
}
=
{
e
,
u
}
whenever
u
2
=
u
, then there exists a linear map
f from
M
n
(
R
)
to
V such that
{
x
,
y
}
=
f
(
x
∘
y
)
,
∀
x
,
y
∈
M
n
(
R
)
. Applying the main result we prove that an invertible linear transformation
θ
on
M
n
(
R
)
preserves idempotent matrices if and only if it is a Jordan automorphism, and a linear transformation
δ
on
M
n
(
R
)
is a Jordan derivation if and only if it is Jordan derivable at all idempotent points. |
---|---|
AbstractList | Let
M
n
(
R
)
be the algebra of all
n
×
n
matrices over a unital commutative ring
R with 2 invertible,
V be an
R-module. It is shown in this article that, if a symmetric bilinear map
{
·
,
·
}
from
M
n
(
R
)
×
M
n
(
R
)
to
V satisfies the condition that
{
u
,
u
}
=
{
e
,
u
}
whenever
u
2
=
u
, then there exists a linear map
f from
M
n
(
R
)
to
V such that
{
x
,
y
}
=
f
(
x
∘
y
)
,
∀
x
,
y
∈
M
n
(
R
)
. Applying the main result we prove that an invertible linear transformation
θ
on
M
n
(
R
)
preserves idempotent matrices if and only if it is a Jordan automorphism, and a linear transformation
δ
on
M
n
(
R
)
is a Jordan derivation if and only if it is Jordan derivable at all idempotent points. |
Author | Ge, Hui Wang, Dengyin Li, Xiaowei |
Author_xml | – sequence: 1 givenname: Dengyin surname: Wang fullname: Wang, Dengyin email: wdengyin@126.com – sequence: 2 givenname: Xiaowei surname: Li fullname: Li, Xiaowei – sequence: 3 givenname: Hui surname: Ge fullname: Ge, Hui |
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CODEN | LAAPAW |
CitedBy_id | crossref_primary_10_1016_j_laa_2011_10_014 crossref_primary_10_1016_j_laa_2012_07_031 crossref_primary_10_1080_03081087_2013_794799 crossref_primary_10_1016_j_laa_2013_09_039 crossref_primary_10_1007_s40840_014_0011_2 crossref_primary_10_1080_03081087_2012_703193 crossref_primary_10_1080_03081087_2013_866668 |
Cites_doi | 10.1016/j.laa.2007.11.018 10.1016/j.jalgebra.2005.11.002 10.1016/S0024-3795(96)00203-0 10.4134/JKMS.2007.44.1.169 10.1016/0024-3795(96)89195-6 10.1016/0024-3795(91)90016-P 10.1080/03081080903191672 10.1016/j.jalgebra.2010.10.037 10.1016/S0024-3795(01)00379-2 10.1016/j.laa.2008.11.011 10.1016/j.laa.2008.04.010 |
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Keywords | Zero product determined algebras Bilinear maps 15A99 15A27 15A04 15A03 Idempotent elements determined algebras Automorphism Matrix algebra Tensor product Commutative ring Idempotent matrix Jordan algebra Commutativity Linear transformation |
Language | English |
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Snippet | Let
M
n
(
R
)
be the algebra of all
n
×
n
matrices over a unital commutative ring
R with 2 invertible,
V be an
R-module. It is shown in this article that, if a... |
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StartPage | 2889 |
SubjectTerms | Algebra Bilinear maps Exact sciences and technology Idempotent elements determined algebras Linear and multilinear algebra, matrix theory Mathematics Sciences and techniques of general use Zero product determined algebras |
Title | Idempotent elements determined matrix algebras |
URI | https://dx.doi.org/10.1016/j.laa.2011.05.002 |
Volume | 435 |
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