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单词 IdealsInMatrixAlgebras
释义

ideals in matrix algebras


Let R be a ring with 1. Consider the ring Mn×n(R) of n×n-matrices with entries taken from R.

It will be shown that there exists a one-to-onecorrespondence between the (two-sided) ideals of R and the(two-sided) ideals of Mn×n(R).

For 1i,jn, let Eij denote the n×n-matrix having entry 1 at position(i,j) and 0 in all other places. It can beeasily checked that

EijEkl={0iffkjEilotherwise.(1)

Let 𝔪 be an ideal in Mn×n(R).

Claim.

The set iR given by

𝔦={xRxis an entryof A𝔪}

is an ideal in R, andm=Mn×n(i).

Proof.

𝔦 since 0𝔦. Now let A=(aij) and B=(bij)be matrices in 𝔪, and x,yR beentries of A and B respectively, sayx=aij and y=bkl. Then the matrix AEjl+EikB𝔪 has x+yat position (i,l), and it follows: If x,y𝔦, then x+y𝔦. Since𝔦 is an ideal in Mn×n(R)it contains, in particular, the matrices DrA and ADr, where

Dr:=i=1nrEii,rR.

thus, rx,xr𝔦. This showsthat 𝔦 is an ideal in R.Furthermore, Mn×n(𝔦)𝔪.

By construction, any matrix A𝔪has entries in 𝔦, so we have

A=1i,jnaijEij,aij𝔦

so Amn×n(𝔦). Therefore𝔪Mn×n(𝔦).∎

A consequence of this is: If F is a field, then Mn×n(F) is simple.

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