# Singular

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 Post subject: How to define an R-order S where S denotes a matrix algebra?Posted: Mon Jun 12, 2017 12:28 am

Joined: Sat Jun 10, 2017 6:40 pm
Posts: 4
Dear Singular Forum,

I have the following question:

Let IC denote the complex numbers.

Set R:=IC[[x]] (power series ring) and denote the one-dimensional R-order [[R, xR, xR],[x^2R, R, x^2R],[x^3R, x^3R, R]] (the latter shall be a matrix) by S.

Unforunately, I was even unable to define this R-order S in Singular, since I only found examples of matrices with fixed entries.

So, my question is:

Is it possible to define and to do calculations with this special R-order S in Singular?

Thanks in advance fo the help.

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 Post subject: Re: How to define an R-order S where S denotes a matrix algebra?Posted: Sat Jun 17, 2017 6:31 pm

Joined: Wed May 25, 2005 4:16 pm
Posts: 214
I do not understand that order: Is it an admissible order on the set of monomials of R? If it is, it can be represented by an integer matrix (according to lemma by Robbiano). If not, it is not usable in the context of Groebner/Standard bases.

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 Post subject: Re: How to define an R-order S where S denotes a matrix algebra?Posted: Sat Jun 17, 2017 8:28 pm

Joined: Sat Jun 10, 2017 6:40 pm
Posts: 4
Hi,

thank you very much for your answer. I should have added some definitions, sorry.

Let R be a Noetherian integral domain with quotient field K.

Let mod(R) denote the category of all finitely generated R-modules.

The torsion-free modules E of mod(R) are called R-lattices. They form a subcategory lat(R).

A non-zero R-algebra S in lat(R) is called R-order.

Unfortunately, I am completely new to the computer algebra system Singular.

I wonder, if it is possible to do calculations with (e.g.) the previously mentioned one-dimensional R-order S in Singular.

I was unable to find examples, but then I noticed the forum.

Kind regards,
Bernhard

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