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http://www.mathematik.uni-kl.de/~zca/Reports_on_ca/02/paper_html/node27.html

What is the algorithm to calculate the kernel of a module as defined in the link above?

According to my understanding, after reading "A Singular Introduction to Commutative Algebra", the kernel is some sort of elimination algorithm, but I don't really understand it.

In the link above, I don't understand $R^r \xrightarrow{A} R^m/Im(B)$ where $A$ and $B$ are polynomial matrices.

In particular, I'm confused about $Im(B)$ which is the image of the zeros (roots) of $B$: substituting a root of $B$ into $B$ makes $B$ zero and then $Im(B)$ would be zero, too.

Thanks for your help.

1 Answers 1

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$Im(B)$ isn't the image of the zeroes (what we usually call the kernel) of $B$, but rather the image of $B$. That is, it is the $R$-submodule of $R^m$ consisting of all $y\in R^m$ such that $y = Bx$ for some $x\in R^s$.

Since $A$ is an $m\times r$ matrix with coefficients in $R$, it defines a homomorphism $R^r\to R^m$ by left-multiplication, as above with $B$. The map $R^r\to R^m/Im(B)$ that they describe is the composition of this map with the quotient map $R^m\to R^m/Im(B)$. Explicitly, it is the map which sends $x\in R^r$ to the coset $Ax + Im(B)\in R^m/Im(B)$.

Thus modulo(A,B), being the kernel of this map, is the set of $x\in R^r$ such that $Ax+Im(B) = Im(B)$, i.e. $Ax\in Im(B)$. That is, it is the preimage of $Im(B)$ under the homomorphism given by $A$, which is probably what they meant to write just after their definition of modulo(A,B).

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    not understand coefficients in R and A*x + Im(B) = A*x+B*x belong to R^m/Im(B) does it mean Ax+Bx = Bx, any example to show the calculation, for example using polynomial matrix,2011-11-23