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Let $A=\mathbb{Q} [x_1,\dots,x_n]$ be the ring of polynomials with rational coefficients. Let $T$ be an $m\times n$ matrix with entries from $A$. Consider it as a morphism of $A$-modules $T\colon A^n\to A^m$.

The kernel of $T$ is a submodule of $A^n$. I heard that there is a notion of Grobner basis of such a submodule.

I will be happy to have a reference where this notion is defined and how to compute it in the above situation.

Remark. The notion of Grobner basis I googled deals with the special case of ideals in $A$.

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    $\begingroup$ Yes. See the book by Ene and Herzog, for example. $\endgroup$
    – Joshua Grochow
    Commented Oct 14, 2022 at 6:29

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What exactly you Googled? There are many standard references, e.g.

"Gröbner bases and primary decomposition of modules", by Elizabeth W.Rutman (https://www.sciencedirect.com/science/article/pii/074771719290019Z)

"Gröbner Bases for the Modules Over Noetherian Polynomial Commutative Rings" by Oswaldo Lezama (https://www.degruyter.com/document/doi/10.1515/GMJ.2008.121/html)

etc.

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