数理解析研究所講究録
1503 巻 2006 年 92-94

92

Endomorphisms of a Module over a Local Ringl

城西大学理学部数学科

石橋宏行 (Hiroyuki Ishibashi)2

Department of Mathematics,

Josai University,
Sakado, Saitama 350-02, JapanThe matrix of an endomorphism of a module $M$ over a ring is completely
is called the matrix ofdetermined by the choice of a basis $X$ for $M$ , where$X$
relative to .
Therefore, it will be natural to seek $X$ giving a simple , which is our primitive
motivation.
Now, let be a field. Then we have a good example of such expressed in a niceform for a suitable $X$ . Indeed, we know the following fact (see$2.1]$ or:
$A$

$R$

$\sigma$$A$$\sigma$

$A$

$A$

$R$

$\mathrm{L}\mathrm{a}\mathrm{n}\mathrm{g}[7,\mathrm{p}557,\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{m}$$\mathrm{H}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{i}\mathrm{n}[4,\mathrm{p}307,\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{m}6.7.1])$

inbe a field. Then there are $m$ elements
Theorem. Let$R[t]$$\{g_{1}(t),g_{2}(t),\cdots,g_{m}(t)\}$overin the polynomial ringand $m$ polynomials$m$ofcompanion
matrices
adirect
sumof
that
is
such
indeterminate
in one$R$$\{x_{1}, x_{2}, \cdots,x_{m}\}$

$R$

$M$

$A$

$t$

$\{g_{1}(t), g_{2}(t), \cdots,g_{m}(t)\}$.

What can we say about this result, if is a local ring ? Is it possible to get aconcise form of as above ? To analize this problem is the purpose of this note.
So, letbe a local ring with the identity 1 and the unique maximal ideal ,$M$ a free module of rankover , and End $RM$ the endomorphism ring of $M$ .
Then we have two chanonical maps
$R$

$A$

$R$

$\mathrm{m}$$n$$\pi_{R}$:$R$$Rarrow\overline{R}=R/\mathrm{m}$and$\pi_{M}$: $Marrow\overline{M}=M/\mathfrak{m}M$

defined bydefined by$a^{\ovalbox{\tt\small REJECT}}\mapsto\overline{a}=a+\mathrm{m}$$xarrow*\overline{x}=x+\mathrm{m}M$

1This is an abstract and the details will be published elsewhere.

2 -maile: hishi@math.josai.ac.jp.

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is a field,
is a vector space over by the scalar multiplication
$a\in
R$
$x\in
M$for
. Clearly the ring homomorphism
andis an -module homomorphism if we define
for $b\in R$ . Alsois an -module homomorphism.
Further, for $x\in M$ and, we obtain an, if we define
endomorphism of , that is,. Thus we have the third chanonicalmap,
by:

Since

$\overline{M}$$\overline{R}$$\overline{R}$$\overline{a}\overline{x}=\overline{ax}$$R$$\pi_{R}$$a\overline{b}=\overline{ab}$

$a,$

$R$

$\pi_{M}$$\sigma\in \mathrm{E}\mathrm{n}\mathrm{d}_{R}M$$\overline{\sigma}\overline{x}=\overline{\sigma x}$$\overline{M}$$\overline{\sigma}$$\overline{\sigma}\in \mathrm{E}\mathrm{n}\mathrm{d}_{\overline{R}}\overline{M}$$\pi_{E}$$\sigma\vdasharrow\overline{\sigma}$$\mathrm{E}\mathrm{n}\mathrm{d}_{R}Marrow \mathrm{E}\mathrm{n}\mathrm{d}_{\overline{R}}\overline{M}$which is a ring homomorphism.
An element
End $RM$ is called a permutation if it is a permutation on some$M$basis for . Alsois diagonal if the matrix of 6 is diagonal relative tosome basis for $M$ .
, and by $M_{f}(R)$ if
Also we denote the ring ofmatrices over by.
Then,
our results are as follows:$r=s$$\rho\in$$\delta\in \mathrm{E}\mathrm{n}\mathrm{d}_{R}M$$R$$r\cross s$Theorem A. For any
on $X$ such that the matrix of$M_{\mathrm{r},s}(R)$there is a new basis $X$ and a permutation
relative to $X$ is expressed as$\sigma\in \mathrm{E}\mathrm{n}\mathrm{d}_{R}M$$\rho^{-1}\sigma$$\rho$

where(i) $m$ is the number of the invariant factors of ,$\overline{\sigma}$(ii) $I_{n-m}\in M_{n-m}(R)$ is the identity matrix,(iii) $O_{n-m,m}\in M_{n-m,m}(R)$ is the zero matrix,(iv) $D_{m}\in(d_{ij})\in M_{m}(R)$ is a matrix withmodulo$\mathrm{m}$$d_{ij}\equiv 0$,

mod$\mathrm{m}$if $i\neq j$ , i.e., diagonaland(v) $B_{m,n-m}=(b_{ij})\in M_{m,n-\mathrm{m}}(R)$ is a matrix such that for any $i=1,2,$

we have$b_{0j}\equiv 0\mathrm{m}\mathrm{o}\mathrm{d} \mathrm{m}$for$j\leq \mathrm{I}\mathrm{I}_{\lambda=1}^{i-1}(n_{\lambda}-1)$

or$\Pi_{\mu=1}^{i}(n_{\mu}-1)<j$.

$\cdots,$$m$

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References[1] J. Dieudonn\’e, Sur les g\’en\’erateurs des groupes classiques, Summa Bras. Math.
3, (1955) 149-178[2] E.W. Ellers, H.Ishibashi, Factorizations of CEIransformations over a Valuation
Ring, Linear Algebra Appl., 85(1987) 17-27.

[3] A.J. Hahn, O.T. O’Meara, The Classical Groups andWissenschaften, vol. 291, Springer-Verlag,$\mathrm{K}$-Theory, Grund. Math.

$\mathrm{B}\mathrm{e}\mathrm{r}\mathrm{l}\mathrm{i}\mathrm{n}/\mathrm{N}\mathrm{e}\mathrm{w}\mathrm{Y}\mathrm{o}\mathrm{r}\mathrm{k}/\mathrm{T}\mathrm{o}\mathrm{k}\mathrm{y}\mathrm{o}$, 1989.

[4] I.N.Herstein, Topics in Algebra (2nd ed.), John Wiley and Sons, New York/Sin-gapore, 1975.

[5] H. Ishibashi, Involutions and Semiinvolutions, Czechoslovak Math. J. (in press)[6] –, Length Problems for Automorphisms of Modules over Local Rings, Linear
Algebla Appl., (to appear).
[7] S. Lang, Algebra (3rd ed.), Addison Wesley, Tokyo, 1999.
[8] B.R. McDonald, Geometric Algebra over Local Rings, Dekker, New York, 1976.