The command
MatrixSpace(R,n,m)
gives all the $n\times m$ matrices over R with a module structure. To construct ideals of matrix rings, I would need something like
MatrixSpace(I,n,m)
where $I$ is an ideal of $R$. But the above command would not work as $I$ is not a ring. Any suggestions what to do? Or more generally, how to construct the set of all matrices with entries from a given set?
First, note n by m matrices over a ring form a ring only if n and m are equal.
Example:
sage: MatrixSpace(ZZ, 2, 3) in Rings
False
sage: MatrixSpace(ZZ, 2) in Rings
True
Now in the case of square matrices, one way to answer your question is as follows.
sage: R.<x> = ZZ[]
sage: I = R.ideal([2, x])
sage: M = MatrixSpace(R, 2)
sage: IM_gens = [a * m for a in I.gens() for m in M.gens()]
sage: IM = M.ideal(IM_gens)
This gives us the following.
sage: R
Univariate Polynomial Ring in x over Integer Ring
sage: I
Ideal (2, x) of Univariate Polynomial Ring in x over Integer Ring
sage: M
Full MatrixSpace of 2 by 2 dense matrices over Univariate Polynomial Ring in x over Integer Ring
sage: IM_gens
[
[2 0] [0 2] [0 0] [0 0] [x 0] [0 x] [0 0] [0 0]
[0 0], [0 0], [2 0], [0 2], [0 0], [0 0], [x 0], [0 x]
]
sage: IM
Twosided Ideal
(
[2 0]
[0 0],
[0 2]
[0 0],
[0 0]
[2 0],
[0 0]
[0 2],
[x 0]
[0 0],
[0 x]
[0 0],
[0 0]
[x 0],
[0 0]
[0 x]
)
of Full MatrixSpace of 2 by 2 dense matrices over Univariate Polynomial Ring in x over Integer Ring
Asked: 2016-12-09 18:09:51 +0100
Seen: 400 times
Last updated: Dec 11 '16
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